You're using a fairly prescriptive definition of race. Race, as commonly used, is a social construct. Ancestry functions the way you describe, but as the example given above shows, "race" and ancestry are not directly related.
Ancestry is objective. Race is subjective. There is a many-to-many relationship between the two. If people say you're black, then you're black. Different groups may come to different conclusions on what your race is depending on varying social norms. If you disagree with this, then you're not using "race" in the same way that the rest of country is.