Hi, Joe, and welcome! I don't know of any particular name for what this girl is up to. But it sounds to me like she really does want a mother to give her what's she's missed out on and wants. And she is jealous of anyone who takes her "mother's" attention away from her. Sad, because it strikes me as more like the behavior of a child.

Is it possible the girl has some type of personality disorder? Time well tell if she gets angry with your girlfriend and then rejects her--only to try to pull her back into the relationship later--a fear of abandonment which suggests borderline personality disorder.

Too bad your girl friend is not able to set boundaries. She must be worried about the girl falling apart, since the girl sounds so needy. Is your girlfriend attracted to those kinds of folks? Does she feel the need for a "daughter"?

You might not know the answers. I am not an expert and am just thinking out loud. I hope your girlfriend will miiss you and rekindle her closeness with you.