In the sense that Vader considers himself a different person from Anakin (one would say "different, yet the same" with emphasis on "different"), does Sidious feel the same about himself in regard to Palpatine, only putting up with being called Palpatine for practical reasons?
This is explicitly answered in the official novelisation for Revenge of the Sith. In short, Sidious considers Palpatine to be his disguise (to the world outside) and Sidious to be his real self.
Then Sidious, for some reason, decided to intervene.
“Don’t fear what you’re feeling, Anakin, use it!” he barked in Palpatine’s voice. “Call upon your fury. Focus it, and he cannot stand against you. Rage is your weapon. Strike now! Strike! Kill
Palpatine examined the damage to his face in a broad expanse of wall mirror. Anakin couldn’t tell if his expression might be revulsion, or if this were merely the new shape of his features. Palpatine lifted one tentative hand to the misshapen horror that he now saw in the mirror, then simply shrugged.
“And so the mask becomes the man,” he sighed with a hint of philosophical melancholy. “I shall miss the face of Palpatine, I think; but for our purpose, the face of Sidious will serve. Yes, it will serve.”
As @hypnosifl has pointed out in a comment, there's a very relevant quote from Palpatine actor Ian McDiarmid that states that Lucas had very much the same idea in mind; Sidious was always the true form and that he wore Palpatine's unblemished face like a mask. When he became hideously scarred, that was, ironically, his true face.
'Naturally, Lucas had a better way of summing up the relationship. "He said this casually, 'You should think of Palpatine's eyes as contact lenses...' So there's Palpatine's eyes and my eyes and that was very interesting. So, in fact, his face, which is the same as mine, was the unreal aspect. My own face was the mask. And then when I get into the mask, that is the evil person - that's the real face."'
Lets consider in what way is Anakin "different" from Vader. While Anakin has some dark attitude he is mostly a light side guy. He is tragically seduced by the dark side and loses all his natural senses and good feelings, murdering children and becoming an emotionless Dark Lord.
At this point, quoting Obi-Wan:
OBI-WAN: When that happened, the good man who was your father was destroyed.
In his redemption, on Return of The Jedi, Vader gets in touch with his deep feelings, starts to care about his son, at that moment he becomes Anakin again, a good man.
Now, when it comes to Palpatine/Sidious:
Sidious is an evil character, he cares about no one but himself. Also he is a master planner, it is all about his plan to become the ultimate ruler of the Universe.
When he is acting as Palpatine, even though he is hiding his dark nature, he is moving forward with his plan and manipulating things to work in his advantage. He is carefully inciting Anakin with the Dark Side, and creating a Galactic war to stay in power.
So, with Anakin/Vader you can see a clear distinction between the two personalities, and there's no point where they coincide.
With Palpatine/Sidious both personalities have the same goal and motivations, and he is acting as both at the same time.
So no, Palpatine is not a different person than Sidious in the same way as Darth Vader is.