The Batman's career as a heroic vigilante working with the Gotham City Police Department was at its peak through the 1960's into the 1980's - at least for Earth Prime-Time (and not necessarily always on screen.)
After that, it was the Toobworld Central conceit that Bruce Wayne retired his secret identity as the "Caped Crusader". And as long as Adam West is alive (if not for longer), then Bruce Wayne continues to reside in Toobworld's Gotham City.
Usually, any mentions in other shows of Bruce Wayne, the Batman, the Batcave, the Bat Signal, etc. could be splained away as being common knowledge, thanks to Bruce Wayne's life story finally going public. Perhaps Wayne published his memoirs which were then adapted for comic books and movies. There may even have been a TV series in Toobworld which starred an actor named Adam West, who bore an uncanny resemblance to Bruce Wayne. (This would be just in case both West and Wayne are referred to in connection to 'Batman'.)
That's how most of the references play out - as pop culture Zonks that can be splained away.
But two new references raise the possibility that the Batman is back on the scene. And most likely there's a new crime-fighter under the cowl.
"Every time Batman goes looking for Justice,
He kills hundreds of people."
"Batman can't catch me"
From a note
left by a serial rapist
'Law & Order: SVU'
Walter Sherman could have been speaking about the Batman of the comics or of the movies, which took liberties with the life story of Bruce Wayne. (It sounds like something that might have happened in a Frank Miller comic story.) Maybe he's been the inspiration for a novel by the Richard Castle of Earth Prime-Time.
As for the line in that rapist's note, Batman was acknowledged to be just as real as the police and the vigilantes and the press, who were all mentioned in the note as well.
Both references treat the Batman as though he's actually still out there fighting crime, just as he was when he was last seen in Toobworld on a regular basis over forty years ago.
I doubt the present day Batman is the original. Bruce Wayne is up in his eighties and everybody already knows about his past life. He'd be easy to track down. And if it was him, he probably killed all those people mentioned by Walter through senile incompetence (NOT incontinence!)
An example of this would be Wayne getting confused and hitting the gas instead of the brakes on the Batmobile. As it is, he probably drives the Batmobile with the left turn signal on all the time.....
Of course we're rejecting that suggestion as it is no way to show respect for the Caped Crusader.
So these two references must be to a new Toobworld character who has assumed the mantle of the Dark Knight. And unlike the straight-arrow Boy Scout whom Bruce Wayne embodied, this Batman is kicking ass and taking names.
So someday the Batman might return to the main Toobworld. And if so, he'll most likely be named Bruce Wayne again. However we could claim that he is the grandson of the original (just in case they decide to do an origin story similar to that told in the comic books.)
By the way, I didn't even consider 'Birds Of Prey' to be part of the main Toobworld. There was too much of that nearly futuristic, slightly dystopian Gotham City that seemed more influenced by the movies. As such, it probably can be found in the "Borderlands" that are adjacent to that part of the Cineverse. Or in the Evil Mirror TV dimension.