B) My father is British, I was born in NJ. I'm both. I'm also Chinese, Scottish, and Irish. Problem, officer?
C) Everyone thinks Americans are idiots, Iraqis are terrorists, Chinese children make iPods on a daily basis, Irish people are always drunk, Russians are Communists, see where I'm going? It's for satirical purposes, really.
There's a difference between having a relative or parent from one nation and being part of that nation.
If you don't live in Britain or don't have a British birth certificate or a Certificate of British Citizenship, then you're not British, especially considering British isn't technically a race, as British covers people from a number of different countries.
If you're an American citizen then you're American. You just happen to have British ancestry in your family. That doesn't make you part of that ancestry nor does it tie you to it.
You're not Chinese, Scottish or Irish. I doubt you're a citizen of any of those countries. Because your long gone family comes from there doesn't mean you do.
The fact that you're mentioning satirical stereotypes doesn't in anyway actually give reason as why it was in anyway funny to claim Americans as being the only people who would get this question wrong.
Not all satire is funny.
Also of note, if you were British, you'd know the correct american stereotype is
Fat and Stupid.
Tip: That's just a stereotype viewed by some. I'm not saying all Americans are such.