Does this work?

I am fairly certain that it does. But just wanted clarification.

As long as you discard a Dark Mirror to a ME AM BIZARRO #1, for example; yes.

Version is much different than Identity/Name.

EDIT: Bizarro World only changes identities, not versions. My bad.

And cards only have the same name if they have the same number of names and all names match. No dice.

If you've got Bizarro World face up, you can't legally use Assorted Aliases with Bizarro anymore.

He's got to have all the same names as the card you discard, which is impossible given that he has ALL names.

question directed to Batshido's post.
Wait..
Assorted Aliases
To play, discard a character card.
Target character gets +3/+3 this attack if it's a different version of the discarded card.

If Biz World is up, I thought it still works, because say you discard a Superman - LSOK to the Bizarro - Dark Mirror in play, Bizaaro still has a different version (LSOK and Dark Mirror?) so shouldn't it work?

No because he has a different name. He has a potentially infinite amount of names. Thusly, any mono named character isn't going to work for effects that trigger off of the same name. No one but another in play bizarro has the name of an in play bizarro.