I'd want to break this into three separate points, which can all stand separately from each other:
1. What's their relationship like in general?
2. Was there a romantic component to it?
3. What does it mean in the narrative?
Part 1: General Relationship Dynamic
So while Rei vaguely exists in Shinji's vicinity since ep 1, he only really gets to know her when he's settled down with friends & all, and notices there's one person in his class who still has no friends. He figures that since they're in the same boat as pilots, they should band together.
However, due to multiple embarrassing accidents, Ramiel's attack, Shinji's awkwardness, and Rei's general closed-off ness, it takes him 2 whole episodes to befriend her. Up until this point Shinji has just gone from one emotional outburst to the next, just reacting to things, concerned with his own fear, bitterness & ambivalence, but now that he's attained a semi-regular life, he is actually trying to do something nice. He sees someone who is all alone, known how much that sucks, and wants to do something about it, even when the others tell him that she's basically the class weirdo.
He has some serious foot in mouth moments like insulting someone she's loyal to or postulating that cause she dutifully does it, she must not have really suffered while piloting (the whole topic of Gendo just sets them both off in different ways) but he's determined to try.
Then the angel attacks, and while Rei doesn't suddenly become all friendly or mince words about the reality of the situation (at all!), she is markedly the one person who doesn't judge Shinji for considering to quit, or try to make him do it. With no pressure to reflexively hold against or use as an excuse, Shinji has to consider what HE actually wants... and turns out that he doesn't want to let her fight the monster alone. (that, and he wants Misato & co to need him, it's not 100% black or white)
(the og series doesn't go into detail about why, but in the Rebuilds its show that since she feel she can't leave or be happy outside EVA, she thinks the ones who can leave & be happy elsewhere might as well do it; - see the elevator scene. She is, after all, expendable, & was created "for" the awful parts - or that's what she thinks at the time.)
Of course during the fight, Shinji saves her, & Rei barely knows how to react to the idea that someone wants to protect her - but from that point on, there's a bond between them - markedly, this is when she goes from calling him just 'you' to using his name (something that maybe went over our heads, but is explicitly noted on the jp wiki - notably when the 'third' shows up in ep 23, she's back to just 'you', up until she betrays Gendo.)
Earlier on, she probably visited the hospital because she's a nice person (she shown similar concern when bad things happen to Asuka and Touji) and she said she'd protect him because it is her job, but now she's come to actually like him.
The thing is that Rei is feeling really alone but she's not fully aware of that, still it shown in her actions that, when someone is nice to her, she really values that. All the more so when here comes someone who, unlike Gendo, has no ulterior motive & doesn't see her as anyone else's bargain basement standin. He just wants to be her friend.
& it more or less continues like that the rest of the show, with Shinji trying as best as he can to awkwardly talk to her & do nice things for her, & Rei in turn grows pretty attached to him to the point that she gets really angry when Asuka insults him while he's MIA.
So, if I had to boil their relationship down to a few bullet points,
- it's very much a "same boat" dynamic - both are socially awkward teens used & exploited by a big scary organization, but at least they can help each other & brave all of this difficult stuff of learning people skills & fighting the angels together
- She's even more introverted than he is, so, if Shinji just passively waited, nothing would happen, so he has to be proactive here. On the other hand, since neither is super loud & talkative, they don't talk over each other either & eventually get fairly comfortable in each other's presence (well, as much as Shinji can chill - but see that clip from ep 22 that implies they have regular talks on their train rides) As a result, you really see their better characteristics in this interaction; it gets them both out of their comfort zones showing their caring sides
- There's also a huge aspect of inspiration - She's very often someone whose example spurs Shinji into action; He admires her strength & determination since he finds it very hard to keep soldiering on
Of course towards the end of the series everything goes downhill, Rei sort of dies & then Shinji just doesn't have the nerve to go talk to the other clone. He is told she's an artificial creation, more or less figured out the Yui thing, he's not even sure in how far she's the same person as the one he knew- He just doesn't know what to do, & has no reliable adults left to even talk this over with.
The segment in EoE implies he's a bit mad she never told him; mostly he's just overwhelmed, doesn't know how to act - even so, he most certainly dropped the ball here & could have been a better friend, after all, if it weirds him out, think hoe disorienting this must be to Rei!
But they more or less each stew in their own issues until the big finale happens.
At this point Rei has lost most of her faith in Gendo & doesn't know if instrumentality is really for the better. So she does what most of us would do: Ask a friend. Alas, she has only one, and he's currently in mid-existential crisis. (which isn't improved by Rei showing merged with lillith)
She had her own reasons too for merging with lillith - she hoped she'd find a place where she belongs, ("I'm home") since that's where she originally came from. Or at least, she hoped her painful existence would finally be over - but Shinji screaming his lungs out up there was the last straw.
Once third impact gets going, she does tell him to like consider it properly & gives arguments against instrumentality ("Then what is your hand for?") but she can't give him all the answers either since she doesn't have it any more figured out...
When Shinji requests one Third Impact, please, tho, she accepts - as for why, look at what her many apparitions say while they tang the JSDF corpses: "The world is full of loneliness and emptiness etc." -her own experience in life was no better than Shinjis.
So, they give it a good try. But Shinji decides it sucks. He still thanks Rei for coming to help him & giving him the option, tho as it helped him sort his issues out.
Part 2: Romantic Component
First I want to point out that friendship is not 'lesser' than romance, nor are the two mutually exclusive. Everything I said above would still be 100% true if they're simply important friends. That said, the answer is definitely yes:
- The proposal (that is, the very first "pitch document" for evangelion) introduces Rei as "...learned about feelings after falling in love"
- In one interview from around the time when EoE was made, Shinji was asked which of the two pilot girls he likes, & the answer was "both... but also neither" (because his issues get in the way to an extent - as Rei said in EoTV, he needs to love himself first...)
- She also brought out an 'inofficial image song' after the last rebuild. The first verse is about Rei, the second about Asuka. Going on about someone's "perfect white arms" is definitely love interest talk.
- One interview relating to the Manga had Sadamoto describing the type of love between them as "love between man & woman" (you can find it on all the Japanese fanwikis since she ship is more popular there) & describing the situation with Kaworu as a "love triangle" - though he flip flops on both their cases depending which interview you look at.
- Various characters like Misato & Touji tease Shinji about having a crush on her, Asuka acts all jealous etc.
- ...her whole death scene where the angel takes her shape & does that weird embrace & she concludes that that's a reflection of her deep desire
- ...just, a whole lot of the cinematography there, the blushes, emphasis on gazes etc.
It's simply what canon depicts, regardless of whether they "should". The whole 'ppl find out they're secretly related' thing is a common tragedy trope as old as books, and in jp media it doesn't even always go hand-in hand with an unhappy ending. It's also not that uncommon in real life for ppl to be ready to get married & then find out one of them is adopted, or a bastard child of the other's parents, or from the same sperm donor etc.
You can't really compare consenting adults (or teens the same age) to, like, some dude raping their daughter or little sister, abusing the preexisting trust to do so & causing lifelong trauma. That sort of thing is bad because its rape, it's exploiting what should be a caregiving bond, & it causes trauma.
None of these things really apply to two people who met as teens and have like a normal classmate relationship. It seems pretty nonsensical that they would somehow think of each other as family if they never lived together. Though it's not an exact match for obvious reasons, even Asuka is almost closer to being a 'sister' because they use first names, live together, act very familiar/casual share parental figures in Misato and Kaji and have some rivalry/spats. Dear only children, take note that a sibling isn't just someone you like but don't boink, it's a specific type of relationship.
Family bonds aren't something that cheap or automatic, they're built over time. If some deadbeat showed up out of nowhere and wanted to start acting like a father, we'd think that is ridiculous. He'd have to get to know the kid & build trust first. So while it is thinkable that, after hearing the revelation, they could eventually build a sibling-like relationship, that would need to be built first.
Stories aren't supposed to be perfect example to imitate, but simply interesting stories, and depictions of feelings.
Part 3: What does it mean
The first thing you've got to understand about Rei is that the point of her is that she is ultimately fundamentally human (there's a really beautiful statement by her VA on this), & that she is her own person, not just a watered-down version of any of the components she was created from.
She was created as a piece of organic hardware, a perfect soldier, an empty vessel to hold Lillith's soul, raised in a sterile, loveless environment with no hope, no dreams, nothing but a cold. clear-cut purpose in the service of someone else, like a mass-produced commodity. No wonder then that she is barely attached to that sort of life. She has little that's truly her own. Not even her face. But under all the numbness & damagedness & the doubt, there's a person with thoughts, feelings, opinions etc if you only bother to pay attention.
She's not a cheap bargain Yui (no matter what Gendo would want), she's just herself.
I really wish ppl who want to pair Shinji with Asuka/kaworu/Mana/Mayumi/Mari/whatevs would just write Shinji & Rei as friends (which they certainly are) instead of erasing her whole struggle & individuality & treat her like the cheap knockoff Yui or lifeless tool that Gendo wants her to be -.-
So, onto the thematic meaning.
Rei as part of a contrast duo
Ovsly EVA is not your usual simple "love triangle story" - rather, the girls, though they each have their own stories, have certain symbolic meanings in regards to Shinji. Sadamoto has described them as "like Ying and Yang" - they're conflicting desires & different drives that act on the human psyches, and Shinji being drawn to both very different girls symbolizes the conflictedness between those disparate drives.
- Asuka - Rei
- Sun - Moon
- extrovert - introvert
- Body/instinct - Spirit/mind
- lust/physical love - spiritual/idealized love
- "Eros" - "thanatos"
- vitality - sterility/ lifelessness
- resistance - resignation,
- defiance - fatalism
- selfish(but also positive self-assertion) - selfless (but also self-abandoning)
- excitement, novelty, risk - calm, security, peaceful
- Pain in interaction - Hope of Understanding
- Misunderstanding - Communication
- pretend - genuine
- tangible, material - ephemeral, ethereal
- life - death
As you see, both have both negative & positive associations. Both were designed to appeal to fundamental parts of human nature, but also human absurdity (There's one interview where Anno says something like "Rei is for the sadists, Asuka is for the masochists" before thating that we all have both tendencies & just vary in their strength. )
Though I don't think Rei fans are generally sadists, my impression is more that extroverts like Asuka & introverts like Rei. But eh, artists are weird
Rei & Shinji as specific themes
While a lot in EVA was left open for interpretation, this is actually one apparently seemed important enough to actually spell it out in EoE:
Shinji: "Who are you, inside my heart?"
Rei: "The hope that people can understand each other"
Kaworu: "The words 'I like you'"
So there you have it: The 'message' is about hope and communication. This is further corroborated by another interview where Anno talks about how he felt he'd written himself into a corner after ep 6 because the two already "fully communicated" in that scene.
You'll notice that in the train/introspection scenes, Rei often gets the part of the 'conscience'. "But did you REALLY try to understand people?"
When Shinji first shows up he's spent his life having super low expectations to protect himself from disappointment, so hope, to him, is something unsettling & distant, kind of like Rei appears at first. Also, at the very early beginning, seeing one person that Gendo is nice to allows Shinji to think that maybe he has a good side after all.
Gradually he comes to have more hope & maybe outright embrace it, then of course everything goes down the drain & he loses it, but some vague ghost of it remains & turns the tide at the end.
Of course the very next line in that scene from EoE is "But that's like an illusion, a kind of prayer, it doesn't last" - they're ephemeral, transient figures, Kaworu & Rei. Rei, in particular, is an ethereal person that you long for but can't quite grasp, since she dies tragically & what not.
Maybe he lost Rei & Kaworu, but from their example, he knows that connection & understanding are possible, & he can find it again. That won't last forever, either, no relationship will be all sunshine & roses without hardship or misunderstanding (that's what the beach scene is all about), but he accepts this at that point because there are good things, too
In that sense Shinji manages to take a very different path from Gendo, who had one person he loved & then when he lost her figured that all is lost & the world can only be fixed through instrumentality.