After ten thousand years, what would remain of any of our heritages? How many here have surnames dating from the Neolithic?
Yes. After ten thousand years I would expect that there would be a completely different set of ethnicities, with their own cultural attributes & baggage and potentially associated phenotypes (if they were physically or socially isolated from each other).
10,000 years ago northern Europeans were hunter-gatherers as dark as south Indians today, and no human adults anywhere could drink cow milk. We all, obviously, speak languages derived ultimately from the languages spoken 10,000 years ago, but no one has yet been able to recover a single word of an actual 10,000 yo language. At even 5000 years ago, within the historical record, the languages are pretty much entirely dead dead dead and the names are completely alien to the modern world.
So using contemporary names as a signpost for ethnicity & culture pinned to the 21C in a setting 10,000 years in the future would force me to suspend my disbelief high enough to kill it. Unless there's an in-story explanation presented up front. (Cf Cordwainer Smith's "Alpha Ralpha Boulevard," a wonderful far-future story that starts out by stating they're having a moment of cultural re-discovery/re-creation of the past and now we're all French.)
"She was not the girl I meant to seek,
I met her by the merest chance.
She did not speak the French of France
But the surded French of Martinique."