On one hand, the person who is asking is being a prick for hammering on a question that the other person clearly doesn't want to answer. On the other hand, it's incredibly obvious that what the person asking wants to know is what race the other person is. The correct way to ask this is "Do you mind if I ask: what race are you?" or "Do you you mind if I ask: are you <race here>". I don't think it's inherently racist or taboo or rude to want to know someone's background.
Am I misunderstanding something? The person did answer the question. She's from the United States. Also, the person doesn't want to know her race, they want to know her nationality and just assume she can't be from America.