Board Thread:General Discussion/@comment-26994636-20161025204929/@comment-27701762-20161025230452

When you fall from the bridge in Waterfall you have a "flashback" to when Chara and Asriel first met. Asriel asks Chara's name, and then responds "Chara, huh?" (or more appropriately, responds with whatever you name Chara). So "Chara" was their name before they fell. There is no indication that they lost their memory. Especially since they remember their hatred for humanity and the Golden Flowers that grew in their village.

Frisk never tells anyone their name because no one asks. The only time anyone asks is at the very end. There's no indication that Frisk lost their memory, and the theory you provide wouldn't make sense, as A) there is no way for Asriel to know Frisk's name before returning that memory to them, unless Frisk had already told him their name, and B) Frisk gives Asriel their name before he returns the souls and their memories.