I learnt something interesting in Cultural Anthropology class. My textbook, Cultural Anthropology by Kottak, mentions that the way we assign race in America is quite exceptional compared to how 'racial' identities are assigned in other countries:
American rules for assigning racial status can be ... arbitrary. In some states, anyone known to have any black ancestor, no matter how remote, is classified as a member of the black race. This is a rule of descent (it assigns social identity on the basis of ancestry), but of a sort that is rare outside the contemporary United States. It is called hypodescent because it automatically places the children of a union or mating between members of different groups in the minority group. Hypodescent helps divide American society into groups that have been unequal in their access to wealth, power, and prestige.
Of course, that is not the only way Americans look at race. There are many who acknowledge or prefer the label 'mixed'. I am of the opinion that a classification system that assigns children to one group is not necessarily worse than a classification system that allows one individual to belong to multiple groups. It all depends on the effects of this classification. People throughout history have always claimed group identities. Before 'race', people identified with the political entity they were subjects of or the place they lived in. E.g. "I am a person of the X Kingdom/Confederation" or "I am so-and-so of Y town." If the 'classification' doesn't result in unequal social access, then it is nothing more than a label.
Most, if not nearly all countries in the world are multiethnic. 'Mixed marriages' happen everywhere, and 'mixed children' are a fact of life and have been so for millenia. And in many traditional societies, there is no real concept of "mixed children" or the derogatory (to some) term "halfbreed" until modern times. That is because children belong to their father's or mother's lineage (depending on whether it is a patrilineal or matrilineal society). So, in the case of a patrilineal society, someone who has a father from Group A and a mother from Group B is a member of Group A, not a half-and-half, at least in theory. From what I've heard from a Korean acquaintance, he saw a Korean friend with an African American mother as Korean, not African American. But it would be different story if the said friend had a Korean mother instead of a Korean father, because the child inherits his father's group membership, not his mother's. in the traditional view.
A common purpose (but not necessarily the only purpose) of gendered descent rules (e.g. matriliny or patriliny) is to determine property inheritance (often tied to family name), location of residence after marriage, and who is considered 'kin'. E.g. traditionally, the Chinese considered even distant cousins of the same patriline off limits as marriage partners, but first cousins from a different patriline (like father's sister's child, or mother's sibling's child) are not off limits since they belong to a different family. In a matrilineal society like the Moxi, property passes from mother to daughter and the children Moxi women have with men of other ethnic groups are considered Moxi. In many patrilineal societies, the son inherits his father's ethnic classification/group membership, family name, and property.
I'm not saying if these systems are 'morally right' or 'wrong' for that matter - does someone from outside the cultures that practice these descent rules have the right to be a moral judge on them? Maybe yes, maybe no. Frankly I'm not comfortable with saying "patrilineal cultures are sexist against women and they should change their inheritance system so that everyone inherits property instead of just the sons/brothers," or "matrilineal cultures are sexist against men and they should change their inheritance system so that everyone inherits property instead of just the daughters." I mean, Americans have very little patience with 'outsiders' criticizing our ways (though we often expect others to comply when we tell them what values to have and how to live their lives), so I'm a little hesitant to criticize other people's customs.
But still, it is interesting to compare the descent rules I've seen in the US to other descent rules elsewhere. Like patriliny/matriliny, hypodescent assigns offspring of a mixed union to one group. But the hypodescent system is somewhat different from these other systems. In the hypodescent system, someone can inherit her/his father's family name, but not his 'race'. Many times, I hear people in the US (and that includes people of Asian, European, African and Native descent) referring to their Eurasian acquaintances born of a European father and Asian mother as 'Asian', but never once did I hear them refer to the same individuals as "white" or "European." It doesn't quite make sense if you think of it - if that individual is both of Asian and European descent, then the label of European should fit just as well, if not better, especially if that person has a European family name. But a Eurasian acquaintance once said to me, "Being half white doesn't make you white." And ironically, children of black and Asian descent are often considered 'black' by society.
The rationale behind hypodescent seems to be the idea that mixed offspring gravitate to the "lower race", or more socially disadvantaged group.