Who are your parents?
What is your experience?
Those are the two major factors. Barack Obama doesn't have 2 black parents, but he's experienced the kind of racism most black men have. The suspicion, the distrust, the unwarranted hostility. So culturally speaking, he has more in common with a regular black American than a carribean or African coming to North America for the first time.
Mariah carey, Rashida Jones, logic, or Blake griffin have most likely never experienced the kind of anti black racism Obama did, despite also being mixed. They are all treated as a type of off-white rather than black and it would be ridiculous for someone like logic to go around talking about "as a black man" and he knows it.