Of course they do not. Malaysians for example are dark colored, as are Indonesians etc. Also Indians are far from pale.
Isn't the opposite also true in Western cultures? In Europe, where I live, people want to look darker, and go sunbathing. And in the US there's also this "fake tan" thing.
It's all about socio-economic status signals.
My understanding is that in British English "Asian" can include Indians and others from around there. I don't know if Russians are included, though.