That is total BS. Not once in this movie did he ever imply that Batman was Bruce Wayne. He knew that there was a relationship between Batman and Rachel, but that was it. Now if he ever started looking into Rachel Dawes' private life, maybe he could have come to that conclusion, but it's not like Bruce is the only man Rachel's ever been with before Harvey. Hell, as far as I know, she was never "with" him at all, other than as a friend. It was even implied in Begins that she had a relationship with D.A. Finch at one time. So yeah, I think even if the Joker started to look into her personal life (I doubt he'd be interested enough to do so), he probably wouldn't find anything out about who Batman is.