NAH, technically there might be case to be made for you being the AH, but I don't know how old you are, or if this is your first relationship, and you seem like you mean well. There is nothing wrong with feeling like you are ill equipped to listen to a particular problem, as you say, you aren't her therapist, and you don't seem to be very comfortable just listening to her, without trying to fix it, so you need to look after your own mental health as well, as it can be exhausting having someone lean on you so heavily. But communication is important in any relationship, and if she feels like she can't share anything with you then learning how to listen to someone might be something worth considering.

There's definitely a time and a place for offering solutions, but the trouble with that is that you are assuming that she doesn't know how to fix it herself. Sometimes we know what it is that we have to do, especially when it comes to mental health, but knowing doesn't make it any easier. Or she might just want reassurance that you care about her regardless of her personality flaws, and think she's amazing despite her insecurities. You don't have to be a mind reader, but you do have to believe het when she tells you what she wants from you, and right now she is telling you that when she is sharing, she does not want you to try to solve it for her. So don't share your opinion unless she asks for it and don't try and rationalise her feelings, because feelings aren't always rational, and that can seem like you are invalidating them.

You could always let her vent, tell her how much whatever she's talking about sucks, and if you feel like you might have a solution that she may not have thought off, say something like "would you like my perspective on this?" if she says yes, then tell her what you think, if she says no then drop it, and go back to listening.

Listening to someone is a skill. It comes easier to some than others, but you can definitely improve on it by trying, and it's a very useful skill to have. I'm sure you could get a lot of good advice on r/relationshipadvice if it was something that you felt comfortable learning at some point.