"Papyrus," any nicknames derived from his name, and any other names he might ever have had, are all very specifically redacted from his recollection. He remembers himself, he remembers other people and their names, but his own name is just gone.
If reminded of his name by someone in game who knows him, he'll hear it, but it won't spark any recollection or familiarity, and it'll fade from his memory again after maybe a few minutes.
Forms of address like 'bro', and his self-declared title of 'THE GREAT --', are still rattling around in his head where he can access them.
New nicknames given by others in the game may stick enough for him to at least recognize them as addressing him - TBD
CONTEXT
Route: Pacifist (post-completed genocide / bargain), with humoring over the puzzles, a hangout rather than a date, plenty of monologues at the human over the phone.
Ending: Frisk doesn't move in with Toriel, Papyrus becomes ambassador. ~12 weeks of time on the surface, involving a number of startled / alarmed reactions to Suddenly, Skeleton, to the information spreading and things growing a little more political about it all. How much he was still effectively mascot vs involved in the decisions, unclear!
Death: The human killed him. To Papyrus's knowledge, he was the first (and hopefully only) one killed.
(Alas, that was not the case - it was the murder spree implied by the photo of this ending:)