I gave examples above. Ethnic group, cultural group, people group. The actual name of the persons country-eg Polish, Korean etc.
White is not a race or a skin colour anyway, nobody has white skin except perhaps someone with albinism. I've met plenty of pale people and most of those were European but many were Asian. (we have a much larger Asian population here in Australia than African) The same way darker skin does not always make someone 'black', we have plenty of dark skinned Asians here as well.
My goodness, you're still not even getting the basics of what race entails, throwing it out entirely without even trying to consider its validity as a descriptor ALONGSIDE ethnicity, nationality, etc. Pretty sure you just made up synonyms after ethnic group, which is really just ethnicity under a different name
White is a race in that it falls under that category of which we use to describe people with lighter skin tone, but you also have Hispanic (which is an ethnicity) people that identify as white in part because of the racial category being tricky (which doesn't make it useless by any means, only that we need to think of it based on socially constructed categories rather than something anthropological or such)
White and black are not used as literally as you claim they are, that's practically common sense, to try and strawman so you can deny race and enforce some idea of color blindness ignores that it still colors people's perspectives, including yours, as much as you may want to claim you don't see it
Asian is considered a separate racial group, we have roughly 5, just to explain: White or Caucasian <the latter not used as much because of unpleasant historical baggage it has> (which technically covers Middle Eastern people, it doesn't preclude them having Arabic ethnicity or such), Black/African America, Asian, American Indian/Alaska Native, Pacific Islander/Native Hawaiian.
And race is not always agreed upon even by anthropologists and sociologists, it doesn't mean it isn't a useful term in the same vein as how taxonomical terms for animals can be confusing, even with modern developments further segmenting into subspecies, etc.
The reason they're called identities is partly because of how we identify with them. You can be a black person and not be black, I'm pretty sure the black community tends to agree that it's about physical appearance and not strictly skin color and white people vary in that intensity as well. And this isn't even getting into the idea that one can affirm multiple racial identities, provided you have that basis in your ancestry, though it can become tricky, given how some races, especially in America, have a lived experience that a white person cannot just affirm versus acknowledging part of their genetic history or such that we might find
So often the problem seems to boil down to, unfortunately, white people that either don't care about their own ancestry or, more commonly, think that identity is so fluid in terms of race that it means nothing when that's not really how it tends to be utilized, especially by scholars who study it extensively as a social construct
Ask black people about this, ask Asian people, ask any racial or ethnic group I don't think you're going to get an easy answer, especially from even a glance at an article that explains the distinction of race and ethnicity in this sense: race is traits you are born with, ethnicity is those that you are raised in and can be as diverse as one's racial identity.
If you, for instance, have a mother that is Indian and a father that is black, like the very likely VP Kamala Harris, and also you have potentially not just her biracial identity, but also specific Indian cultures or black traditions, such as her father's Jamaican roots and her mother's specific background from Chennai India (which I don't believe I know much about, though there is an episode of Postcards From Buster which looks at it that I vaguely recall watching, along with several others that really opened my perspective on various religious and ethnic groups, like the Kurds)