Just call him the Joker

Throughout the years, the character has evolved, literally in all directions. He was both an enforcer, as well as an immortal. He was a regular man (as regular as such a character can get) as well as a nigh-omnipotent being. But even if any of those things changed, he was always the same.

Even when he was cured of his condition, you could have still recognized him, because of his hair, because of his smile, or because of his clothing preferences.

For this reason, the character cannot be impersonated by any other DC villain. Just like there is only one Batman, in the same way there is only one Joker – both of their names begin with The, and both of them are highly recognizable, no matter what aliases they use.