The wiki page for dissociative identity disorder says “the diagnosis requires that at least two personalities routinely take control of the individual’s behavior with an associated memory loss that goes beyond normal forgetfulness”. What would be the term for if instead the patient thought they were living two different lives simultaneously?
I’m thinking as an example, the Doctor that wrote a book of thinly disguised patient stories, some amalgams - one was a scientist working on an important government project, he was treating because he was convinced he was leading simultaneous lives, his regular one and also one as some kind of space adventurer. Supposedly he eventually cures him, and the man goes on to write his adventures down as a popular sci-fi series of novels.