TBH, i had this argument over there, i lived there briefly due to a work contract, and had this argument with them, i questioned the term "African American" ,i said "you are either American, or African, but there is no such thing as an African American" to which i was told yes there is, and said "Well, you might be American and of African Heritage etc. but you quite literally cant be born in two places". My partner is British, but has dual citizenship and has lived here for 20 years, but she isnt "Brittish Australian"
Another thing that got up my hump was when people heard my accent (its fairly strong, think Billy Connolly mixed with Ewan McGregor and your kinda there) and said where are you from? "Scotland" said me "Glasgow to be more exact" (not that anyone knew where the biggest city in Scotland was) and some people said to me "oh yeah, i am Scottish" or they said "I am Scots/Irish" and would question it "How so? where were you born?" i would ask "Oh i was born in the USA but my great, great, great Grandpappy was from Scotland and my great great Gandma was Irish, so i am Scots/Irish". "No you're not" would be the reply "You're an American"
that never went down well.