My two cents: You should be able to create a player character who looks like absolutely anything. I don't understand people who just want to play a character who looks like them, but that might be because I'm a white guy. I play women around half the time (usually if a game has third-person cutscenes I prefer playing a woman, if strictly first person I prefer playing a guy). When it comes to race...it depends upon context. When I play fantasy games I typically play "white" character, because the settings are based upon European folklore, more or less. But when I play something like Mass Effect, my characters are always multiracial, because I feel like there would be very few people who aren't multiracial in the far future.
I will admit however that I have not liked the trend in modern fantasy (both games and movies/TV) to just pepper people of color around without really any context/explanation. The modern era - with people of very different looks all jumbled together - is a historical aberration. Traditionally speaking if a society is not racist, people intermix freely with one another, and in a few generations everybody looks pretty much the same again. I do think there needs to be more diversity in fantasy, but I'd rather this happen through say settings where the majority of characters are black, or big cosmopolitan cities. Going to random small villages and just finding a black dude or an Asian lady there with no explanation just...destroys immersion for me. It makes me realize I'm consuming media instead of being swept up in a story.