[gotostatement]

The point in saying "race is a socially constructed category" is to counter the ideology that says that "race is a meaningful categorization of people" and, in particular, "it says meaningful things about people's nature and abilities." While ethnicity does pass along, the traits that it carries with it are so scattered and superficial that it's not meaningful in most contexts to categorize people as such, genetically speaking (though of course they can become unified by how they are treated in society).

Analogy: having brown hair is a genetic trait. But it wouldn't make sense if we categorized people as "brown-hairs" because it's just not a meaningful category that says anything much about anything except the color of their hair.