Not necessarily. As an anecdote, I don’t think of myself as a person with a specific name. If I look at the mirror and try to say my name it just does not sound like it fits the person in the mirror. Also I can tell different names to baristas in Starbucks etc. It’s not that I dislike my name either, I think it’s a great name overall.
I never gave it particular thought so maybe I’m in the tiny minority though.