With an imitation of this question: Does Sidious consider himself separate from Palpatine?
Inspired by these two questions: Do the Jedi and the public know that Count Dooku is also Darth Tyranus? and Why did Vader and Maul abandon their name while the other Sith Lords in the Movie didn't?.
Does Tyranus feel the same about himself to Dooku, only putting up with being called Dooku for some reasons? Unlike Vader, who feels that his former identity Anakin is different.