Since a person’s race is only identifiable insofar as other people perceive it, that perception must be tied up intimately with what one categorizes as a “race” to begin with. This is why “race” is a separate term from “ethnicity,” which is more immutable.
Obviously a person’s skin color does not change on the way from Africa to America. But his “race” certainly does. To pick up on Robert’s example: Kenyans are nearly all what Americans would call “black”—that is a single, very large racial category in America. But to say that Kenyans would identify all other Kenyans to be of the same “black” race is, should you ever go there and try it out, an argument that will find little traction. And I am not referring to ethnic divides, either; some Kenyans are perceived by each other as darker and some lighter (even if they are not, to my eyes, lighter or darker) frequently depending on where they are from. But this distinction that is important to Kenyans is obliterated when a Kenyan moves to a more heterogeneous America.
Race can mean, per the definition Robert provides, that one is from the same “tribe,” but it can also mean one is from the same “nation,” and the level of specificity in that categorization entirely depends on where you are in the world. This is the whole point of the “social construction of race.” It is impossible to separate one’s race from another’s perception of racial difference.
I agree with Robert that the claim that ethnicity is not genetic is bogus. I’m not sure how they could think otherwise now that DNA tests have so effectively located African-Americans’ often highly localized African ancestries. (Ethnicity can still be socially constructed, however, insofar as it is a category people put themselves and others into in human society. For instance, if someone were to claim he were “Bantu,” he may be correct but simply imprecise, since Bantu contains with it many other, more specific ethnicities.)