I just wanted to include an example of one of my experiences of this as well.

My first encounter with this type of situation, I had just introduced myself to my (white) patient and her middle-aged daughter and let them know that I would be helping to take care of them that day. The first thing the daughter asked me was "where are you from" to which I replied "I'm from (the city I'm currently living in)". She then pressed on to say, "No, where are you FROM. You know, your race? Where your parents are from?". I was a bit taken aback by how insistent she was and kind of paused and went "Oh, uhhh I'm-" and was about to tell her my ethnicity, when she interrupted me and seemed like she was 'defending' herself and her questioning. She said "I'm allowed to ask this, there's nothing wrong with me asking this. I'm allowed to know this, this is my right. Just tell me your race." So I mumbled "I'm Korean", because at that point I was kind of disheartened and didn't really know what just happened. The daughter replied "Ok good" and I left the room. But that left me thinking about this for the rest of the day.

But how am I supposed to respond in this kind of situation?