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.
Sith are a little ...unhinged :) , but to claim that Count Dooku (one of more sane Sith Lords, with clear ideological goal) had split personality disorder is a bit of overstretch :)
On serious note, whole Banite Order of Sith Lords operated on principle of secrecy. Beside Master and Apprentice, very few people had enough information to connect Sith identity with real public identity of Sith. Dooku was not exception of this rule, only true Sith (if you consider Maul as just an assassin) that revealed his Sith name was Darth Vader. And he had specific reason for that - trying to hide and kill his former self.
As for the question which identity was more important to Dooku, IMHO he was proud Serenno noble to the bitter end. His Sith personality and membership in Sith Order was just a tool to achieve his political goals.
