Sounds like a Ship of Theseus problem.
If Bruce Wayne 'died' the same night his parents and Batman was the real guy then who was he between the period after his parents death and the bat crashing through his window? The latter event is what prompted the idea of becoming Batman. Him falling into the cave at six may have seeded the bat imagery in his mind but bat crashing through his window is what inspired him to create the suit. The only logical explanation is that he was always Bruce Wayne and that the Batman identity is a shell, a coping mechanism and an instrument Bruce Wayne uses to deal with the tragedy that robbed him of his parents and to ensure nobody else goes through what he did.
For Clark Kent, he was raised by Martha and Johnathan, he grew up in the US all his life, his values, outlook on life and perspective were all shaped by his parents. By all accounts he is a descent man but with God like abilities. His powers make him 'alien'. Yet if he were to display those same powers on Krypton in front of an average Kryptonian, they would be just as in awe of him as any human would be. Thus his powers aren't his connection to his heritage. Wait, this is getting off topic.
I think this debate would be more interesting with characters like Wonder Woman, Green Arrow and Black Canary. These characters have had some wild changes over the years. WW seems to get one almost ever decade or so with mixed reception. The Zeus origin did not go down well. Changing one key element flips your entire perspective on the character. The Green Arrow of today is almost unrecognizable from his Golden Age counterpart. His personality has changed from the GA Batman rip off to Denny O'Neil's bearded hotheaded liberal to Mike Grell's lethal vigilante to Dixon's play boy to the Nu52 version to Jeff Lemire's take on the character then you have adaptations like Smallville and the Arroweverse which only have the barest similarities to Green Arrow of the comics but are still popular nonetheless. Black Canary, similarly have undergone multiple changes.