Before the advent of motorized transport, an officer's batman was also in charge of the officer's "bat-horse" that carried the officer's kit during a campaign.
It's quite likely that Alfred was a Batman before Batman was.
Batman and Dracula are characters that share a nocturnal bat related identity and are also wealthy recluses with mansions. So yeah, that's pretty much the joke. Whoops Harley and the Joker got them confused.