What you really want with a categorization scheme is to reduce variance. You ask, "how much variance reduction do I see using this categorization scheme?" This could be expressed as a percentage of total variance, like "U.S. Census 5-races captures 27% of variance".

Race is real. U.S. race categories are not the same as used elsewhere in the world. U.S. Census categories map quite well to genetics. They capture a lot of variance. Ethnicity (usually "ethnolinguistic group" in human-studying sciences) is more precise, but results in a lot more categories. You can capture all the variance with infinite categories. This is the bias-variance tradeoff. This is how things are.