AFAIK he has plenty of fangirls in Japan & fangirls are a thing in general especially with anime. See for example the amount of fangirl lust our very own Kaworu engenders.
Though Shinji strikes me as the sort of character that you appreciate more from a first or third person perspective (finding him relatable or an interesting character concept) rather than in the second person "They're so awesome I'd like to adopt/befriend/date them" sort. Not that he doesn't have potential at all.
See this artwork highlighting his butt:
All things considered he's pretty nice, sensitive, vaguely artsy and
technically a fearsome prodigy mecha pilot, especially in a day-to-day situation where he wasn't just put through the wringer, but if he evaded the wringer completely, where would the plot be?
I don't think it's just his being controversial. Asuka & Rei have plenty of haters too but that doesn't stop the fanbois/ the amounts more or less balance out.
When it comes to Shinji, the show itself gives us a lot of 1st person POV on him generally highlighting his maturity shortcomings, relative youth & showing some very... "unidealized" parts of his. By contrast we see Rei, Asuka & Kaworu through the PoV of someone who, though the overall matter is more complex, distinctly has the hots for them, though all of their characters (except possibly Kaworu) also have their "messy, hairy, ugly" parts to them.