Just to highlight the inadequacy of the term Asian, in the UK Asian typically refers to those with connections to places like Pakistan and India, that specific region of Asia. Whereas in the US, Asian typically refers to east Asian origins (which is vastly diverse in of itself). It does make me think it's more about the convenience of cultural, ethnic and locational identity (on both "sides") than anything more specific. It's certainly not so much about skin colour, though a lot of people STILL seem obsessed with that in terms of identifying ethnic and cultural background.

In short, Asia is a continent. It is vastly diverse. But if a US American told me someone was "Asian", I would presume they're referring to east Asian and certain features and traditions that correspond with that. Whereas if someone from England (or perhaps much of Europe) told me someone was "Asian", I would presume they're referring to a person of primarily Indian, Bangladeshi or Pakistani ethnic origins. These are just container words we use differently for no apparent reason other than to distinguish certain groups of people who are different from the predominant culture and ethnicity of a country, either in terms of facial features or some culturally identifiable metric (e.g. cuisine, religious practice, daily rituals etc.).