Think maybe a first name isn't enough to make the connection. Well Batman was kind enough to give his arch-enemy his victims' last name as well. The Joker doesn't particularly care about Bruce Wayne just Batman. He might be insane but he's smart. He can put the pieces together.
The name is one he remember very well. In fact he used it to gloat Nightwing on during a fight. (Which briefly resulted in him killing the clown.) I kind of miss the Bat family getting this defensive over Jason. Even if Batman was incredibly reckless.