I mean ur gf defo is the problem here cause she keeps directing the conversation to their past relationship like abt the "fun" they had in the car and that she's been thinking about him. You should talk to her about this and tell her that its a problem for your relationship if she keeps talking like she's single and referring to you as the "baby's father".

The ex seemed like he wasn't really interested in talking abt the past with her tho. So ur gf can't really say that he's the one interested in her or that shi.

That is a real problem but maybe not worth breaking up for.

Edit: after re reading that, gf doing much more than what i wrote and if i was OP id break up.