He did not like hearing people separate himself from his future actions. He felt it severed him from his responsibilities for those future actions. Anakin Skywalker would choose to join Palpatine, embrace the Dark Side and fall, and then once there he would choose to commit many terrible acts that could not be numbered there would be so many. Vader was the new name he would use, but the person behind each name was the same person, just at one point in his life he would choose to do evil.
From many long arguments, Anakin knew Obi-Wan liked to think of them as two separate, distinct people, he suspected doing so just made it easier for people to reconcile what would happen. But to him, it wasn’t the truth. He would be Vader, Vader would be him. And separating him from Vader denied the existence of the part of him that could come back.
Not that once he’d reached the point of his confrontation with Obi-Wan there would be much to come back to. It was too late at that point. But it was still him.
“I don’t consider myself separate from Vader,” he said. “By now, I am accustomed to hearing how much people hate me. And that was expected of you, that’s how Sith are, to become one officially you need to assume the place of one of the two already existing. You can wait for one to die, or kill one at take their spot. It’s how apprentices become Masters, and how Masters weed out the apprentices that simply don’t meet their requirements. Your loathing of me was all part of Palpatine’s manipulation.”