You are correct in saying that asian isn't a race, but it is a useful general term because people from asia are asian.
British people are all british, but individually are English, Scottish, Irish, or Welsh (for the most part, not sure what the people in the surrounding islands count as)
British people are overall European, which includes most, if not all of Eurode (Spain, France, Germany, Poland, Sweeden, ect)
The USA is also a diverse country, but most who live there tend to still be North american. And north americans tend to be from The US, Canada, or Mexico.
Its a non issue. If being called Asian, when you are from china, India, or any other country in Asia offends you, then its your own problem.