Comparing a static piece of art with an interactive video game where in many cases your avatar is supposed to represent you is incredibly off-base. I can't imagine games like Mass Effect not having the ability to change things like race and gender (and even sexuality through in-game choices). In an extreme example, imagine something like World of Warcraft with no female characters. Literally millions of dudes and not a single female character.
Edit: Note that I don't think the main characters in AC games are necessarily supposed to be representations of the player. My response is directed at the comparison between games and paintings.
I'd say that it really depends on the game, in WoW you name that character, you customise their appearance, you interact with others as that character. That character is you, it pretty much has no personality beyond your own.
In AC, the devs are telling the story of a character who has his own personality,who has to do a series of set actions within the games set plot, who says thinks based off of a predefined script with a predefined ending. You control the character, you might be able to relate to that character, but that character isnt you.
Really, a game like AC is meant to tell you a story, make you feel and have certain emotional responses, very much like a painting. You start to tell the artist what he can & cant do and you ruin the message he or she is trying to convey.
As for Mass Effect, they've gone for a sort of hybrid of both styles, thats great. But that isnt how Ubi decided to make AC, thats their choice. And anyway, if we werent used to character customisation in ME, would we really notice it if it wasnt in there? I doubt it.