He knows (usually, but his mind wanders) that Nightwing used to be Robin, if for no other reason than one of the first things Nightwing did was visit Arkham to tell the Joker that he was back in business.

He probably does know Batman is Bruce Wayne. He knows that he killed a Robin whose first name was Jason (Batman kept shrieking his name out during a particularly bad moment of "Knightfall") and he'd have to be stupid (rather than mad) to not realise that the Robin he killed in Ethiopia died at the same time in the same place as Bruce Wayne's ward Jason Todd.

Whether he remembers consistently, much less cares, is less certain.