The Last Airbender is a problem because even in the animated style, most of the characters are clearly not caucasian. And there was that great image that went around about how Aang is actually drawn very Asian, but looks "white" to people with a terrible stereotype about Asian appearance.
With anime, it's an old issue that some folks don't know the full story behind. Anime and manga designers started using what is taken to be the "japanese" style to differentiate their characters from one another in a large cast, and manga being mostly black and white caused them to design characters who read best with light complexions.
Plus as observed upthread, anime/manga is simplified and we all read our own race and face into simplified representations. There are a number of common anime facial archtypes that are clearly not caucasian even if their skin is portrayed as being light. The issue is further clouded in that in contemporary times, many Japanese character designers try to blend features of many races together, so that their characters may have international appeal.
But there's also this classic video:
http://www.youtube.com/watch?v=JKTvFhRbBt8
One of the more interesting examples, I thought, where the photorealistic yet stylized characters in the Final Fantasy VII Advent Children movie. There was interesting racial blending going on there, where the primary cast of light skinned characters are not entirely western, but not entirely Asian either.
http://i.imgur.com/37Xdz.jpg
http://i.imgur.com/7Xzr5l.jpg
Seriously, these are not archetypical white people. (Talking about
Reno in the second image, lol.) If most anime characters were rendered photo-real, this is more what the designers would be imagining.