The way that I interpret the pacifist ending is that after Frisk/Chara restores everyone's memories by "reaching for their souls", he/she then finds his/her own soul (which Asgore would have taken when he/she died) and recovers his/her own memories. Chara now fully realizes that he/she is the same person as the first human. After the Asrial fight, Chara sees that Asrial is coming to terms with grief. Because of what had happened before, where Asrial put everyone's lives in danger, Chara does not want to reveal who he/she is, as it would freak everyone out and make Asrial return to his obsessive self. Chara decides to use the fake name "Frisk", to protect everyone.
I keep seeing people say that Frisk and Chara are different people, but I don't see how that would work (especially considering that Chara will keep whatever name you enter, when you do a genocide run). I see various theories about the game which state this like a fact. People who do this without giving any evidence can be a bit annoying.
What evidence is there to justify the theory that Chara and Frisk are different people?