I don't buy the urban myth thing and I never have.
If Batman were real, he'd be just as much a part of pop culture as he is as a fictional character in the real world. There would be Bat-women, Bat-dogs, Bat-kids, Bat-boys, Bat-maniacs, Bat-mites et cetera, around everywhere, mostly as living jokes, not serious crimefighters. And there's not a damn thing Batman could do about it.