"Is he the original Batman?"
"No way, he'd be a hundred years old!"
"Maybe he's immortal..."
The Dark Knight had an interesting take on this with the impostor Batmen in the beginning using guns, and the real Batman showing up to stop them. (There's something to be said about culture and intellectual property in there too, but that's for another day). I'm sure there are plenty of other good examples as well, but my knowledge of superheros is rather sparse as I was unfortunately never that big into comics.
So you got a number of the main points right.