I apologize if this point is too obvious, but the reason these odd situations occur is that many people mistakenly believe that categories such as "white" and "black" correspond to intrinsic, biological categories, that it makes sense to ask if someone is "really" of one race or another. Whereas racial categories like these are socially constructed, and do not correspond all that well with genetic population groupings. The subject of the story categorizes himself as "black" because he categorizes his parents as "black". But they were Americans with very light skin, which means that their ancestry originated from both Europe and Africa, the same as all "white" and "black" Americans, and this categorization is a social convention. The subject's classifying himself as "black" is arbitrary, a remnant of the racist "one drop" attitude that helped support slavery and, later, segregation. The book referred to in the article, Black Like Me, is great reading, by the way.

EDIT: Some replies suggest that I may have been unclear. The social grouping into race is obviously not random, but based on perception of visible morphological attributes. However the categorization that results is probably not the same categorization that you would get if you looked at the whole genotype and tried grouping humans together based on genetics. One basic fact of human genetics should guide you here: there is more genetic variability within any socially constructed "race" than between "racial" groups. So claims that common racial categories correspond to big differences in genotypes don't make sense. Whether there are races at all in humans is something that biologists disagree about, with most at the moment saying "no". Whether race is useful to medical diagnosis is also contentious. The human species is unusually genetically uniform. Most animals that all look about the same to us, like chimpanzees, have (in that case) about three times our genetic diversity.