It's actually an idea that's tossed around now and again by more imagative writers. Some writers suggest, as it was in Death in the Family and other stories, that in fact the Joker does know who Bruce Wayne is. Most writers, IMO, would say that if the Joker knew, he would lash out directly at Bruce Wayne, perhaps even killing someone like Alfred.
But if you try to put yourself in Joker's demented head, you can also see that question in a way that some other writers do. For The Joker, the mask IS the reality. Batman, in costume, on a rooftop, is the real man. Whoever he is underneath is a mere fiction. And Joker is only interested in defeating the real deal - Batman. Not who he pretends to be during the day.
As a result, the Joker honestly doesn't care who Batman is. Even if he did know, it wouldn't change anything. But if he does, in the warped logic through which he defines the world, he just wouldn't care.