If somebody asked me, "What's your name?"
and I replied, "My name is George. How about you?" would it be idiomatic?
Or should I say, "My name is George. How about yours?"
Or "My name is George. What about you/yours?"
Neither Brits nor Americans need to ask, because usually as soon as we introduce ourselves the other person will say hello and tell you their name. If we need to hear it again we'll ask them, but not using How about your name?