A lot of the Final Fantasy games have characters that are impossible to customize (aside from their names and their gear).Continuity said:"In Games where you can customize the character..."
Otherwise known as role-playing games or RPGs.
Aside from that there are plenty of FPS titles out there where you can customize your character's looks and name (inventing custom "skins" and choosing from a wide variety of skins have been a common thing even since the first Quake came out).
Also, speaking as a person who have played REAL roleplaying games for years (i.e the tabletop kind where most of the action takes place in the imagination of the players) I must profess that a lot of videogames claiming to be "roleplaying games" aren't about roleplaying at all. Especially not most MMORPG's which are basically just about looting and hack and slash fighting.
Just sayin'