Now to clarify, I only mean games where you can make your own character, not narratives written for multiple characters.
And the problem I think is mostly due to the immense complexities it would create. Worlds that have to accomodate all sorts of character can not make much of a statement regarding the differences within these characters. At least to me all the Bioware games I played and also Skyrim and other RPGs take me out of the immersion due to the vague or incongruous interactions the world has with my character (admittedly the bioware games are further brought down in what they can achieve with the moral choice systems that they need to connect to my character as well).
For example, take The Last of Us, I think this would have seemed somewhat odd if instead of a Joel I would have been able to choose a small, black lady with a funny hat. At the very least I would have wondered why my funny hat isn't always the only topic of discussion...
Ultimately I think that, at least with the limitations we have currently, can really only say literally nothing about my choice of character and how it relates to the world (perhaps by making the world very bland and/or unrealistic) or at least have moments of "wait...that doesn't make sense, really).
Again, I say the narrative suffers, as in, you could make a more powerful story if you didn't have to account for these choices, not that they are all shit. Both Skyrim and Mass Effect have very good narratives and it works mostly.