Does Joker know who Batman is?
Short answer: It doesn't matter.
Long answer: He might, he might not, but in Joker's mind it's much more fun to interact with Batman; interacting with the man behind the mask is boring.
A lot of Joker's motivation is explained in the answer to this question: What did the Joker want to prove in The Dark Knight?
To paraphrase: Joker wants chaos. Threatening to kill people if Batman doesn't reveal himself isn't motivated by Joker wanting to know who Batman is, it's motivated by wanting to cause panic and chaos.
This is also why he stops Mr. Reese from revealing Batman's identity; Joker really doesn't want to know.