I'm not so sure I agree with this. Growing up as a half black person raised by my white mother and with no knowledge of the "black" half of my family, it's definitely news to me that I'm only black because I choose to identify that way. I guess it's just coincidence that everyone else in my surroundings chooses to identify me as black? I certainly don't feel that I ever made the choice to be.
Everyone else chose to identify you as black, but is that how you identify yourself? I guess when you can be identified physically as "probably" belonging to one group then that is what others will label you as. I think my argument breaks down when the culture crosses racial lines as it becomes possible for strangers to place you in one box or another on sight without interacting with you. As you say, you wont be afforded a choice in the matter. I guess my main experience in this is the division between religious groups here in Ireland where you cannot tell by sight or accent (at least not 100%) what culture a person belongs too. In the the recent past being part of the wrong religious group in the wrong part of the country could get you killed. My argument that culture is something you identify with is flawed in many way but there is a grain of truth in there trying to hide.