Actually, skin pigmentation can be one of the criteria that determines to which race a person is considered to belong. My general biology students are always surprised to learn that race is a term that has no biological basis. It is a term that is best described in a study of sociology rather than biology. Biologically speaking, all of us are members of the "human" race and are members of the same biological species, Homo sapiens. The biological definition of a species is very simple and very specific. Members of a species can produce viable offspring which can in turn produce viable offspring. The definition of race is not nearly so simple and specific!
My take is that these days the PC definition of race has been reduced to skin color.
Even worse is the lack of any pretense of nuance. There are but two bins; White and of Color. But there is no objective measure to tell where the border of the bins lie so there is really no way to argue with those making the assignments of people into bins.