Of course fish are not vegetables; but language is defined by usage and not logic - whether you like it or not, many people use the word "vegetarian" to mean "non meat-eater" and there's nothing you can do about that. You could try using "pesce-vegetarian" or "pescetarian" yourself and hope that people catch on.
I think when Rei says she dislikes meat, by extension that would of course mean fish too.
I would not assume that, but simply ask whether or not she ate fish.