But that's a hard lesson to learn, especially in a society where physical differences have been used to enforce cultural difference and thus have come to be an external sign of cultural difference. This is from a New York Times op-ed by a white man who's adopted black children:
In the case of transracial adoption, there is the force of horizontal identity, where the child looks for others with the same experience of being adopted, but the vertical identity is complicated as well. When we wake them up in the morning, our kids don’t see parents who look like them. For many young transracial adoptees, every time they look in the mirror it’s a shock to see that they are black or Asian and not white like their parents....It’s not just physical; the cultural vertical identity is at stake as well. By adopting children of another race or ethnicity we cut them off from the world they were born in. The National Association of Black Social Workers declared in a resolution in 1972 that transracial adoption was cultural genocide. The wording was, and is, cruel, but it is hard not to see its deeper truth: a Korean or black kid raised in a white world has lost his or her culture.
In what sense is that so? The only culture you have is the one (or ones) in which you are raised and which you have subsequently assimilated.
In most ordinary circumstances, yes, you will inherit your cultural identity from your biological parents simply because you live with your biological parents. A black child who is separated from her parents at birth and is then raised by a white family (or a Chinese family for that matter) will be raised in the culture of that family. That WILL be the child's family. That the child and her parents are not racially the same is irrelevant to the matter of cultural identity.
That people will treat the child as though he should be culturally different from her parents is unfortunate and it does have practical consequences. But that does not change the fact that cultural practices are not inherited in the same way as skin color or hair texture.
Things, of course, are more complicated when a child is partially raised in one family but then adopted by another. But that's just as true of a 6 year old Kenyan child adopted by a black family in Alabama and a 6 year old Spanish child adopted by a white family in Moscow as it is of a 6 year old black child from Alabama being adopted by a white family in Maine.