Question about Batgirl from a non-comics-reader

Okay, to start off: I'm not a big comics guy, so this may come across as a "dumb" question to you guys.

Batgirl is Commissioner Gordon's daughter Barbara. That much is usually certain. Batman does not know this, at least in some iterations of the story. So if Batman doesn't know who Batgirl is, and even if he does know but didn't invite her along to be part of his crime-fighting venture, why is he (and everyone else) okay with some stranger just showing up and ripping off his style?

If she's going to go fight crime in a costume, she could have created any identity in the world. She could have been "The Librarian" or something. But she takes the whole bat motif and runs with it, and everyone's just cool with that?

Or was this dealt with somewhere?

Just curious, thanks.