
Joker is the only person that truly understands Batman.
I forgot where it was from, but there was this part where a bunch of Arkham Asylum (?) inmates are gathered around an unconscious Batman, and the Joker walks to them to ask what they are doing. They reply that they are going to take off Batman's mask to see who Batman really is, and the Joker replies: "Why? This is who he truly is. This is his true face" (paraphrasing).
I wish I could remember where that was from.
I agree. To the Joker, Batman is just Batman. He doesn't care about whose face is under the mask, because to him Batman is his true identity.
Which is true. Bruce Wayne is just Batman's alter ego.