You're right - why would a naming convention continue for 1000+ years...oh wait, you're WRONG:Cleveland Brooker... way to start, Disco Season 3. Because, yeah, in 1000 years we can expect people to have still XX century names and last names. Wow, just WOW. I have been questioning Discovery writers since season 1, and things are just falling off the cliff.
You know, going 1k years into the future is risky for the franchise. First, you risk making all the past canon irrelevant. And second, you would expect things to be so drastically different, that depiction of this alien future could fail easily. Why didn't they pick a more reasonable 2-300 years into the future? Think about how things are different now compared to the XI century. Culturally, visually, scientifically... Plus, they are now messing with a period of time already touched by ENT with the Temporal Cold War. I can honestly see this going very bad, very quickly. Lots of downside, very little upside.
What made TNG so successful with fans was, IMO, that while in the future it wasn't that far off from TOS to be unrecognizable. 100 years are a good time frame where lots of change can happen, but everything can still be tied back to the old stories.
Now, 1k years is a HUGE gap. All we know could well be irrelevant, but the gap is so large that so many things could have happened that all that happened in the past lost relevance. Like you were living in the Roman Empire, and next thing you now you are in the USSR. You lose completely context and relevance... UNLESS... the past is somewhat used as foundation for the present (i.e., Roman and Greek civilization were the foundation of modern Western Society).
If in the far future the Federation didn't exist anymore (or the ideals/concept inspired by them), then Discovery Season 3 would totally be a waste of everyone's time and another bad attempt to reboot the franchise. Which, honestly, has been underwhelming since the beginning, plagued by rushed screenplay and very poor character development.
I am as excited about Picard and Lower Decks as I am totally not looking forward to Disco Season 3.
https://en.wikipedia.org/wiki/Surname
One small excert as an example:
***In China, according to legend, family names started with Emperor Fu Xi in 2852 BC.[10][11] His administration standardised the naming system in order to facilitate census-taking, and the use of census information. Originally, Chinese surnames were derived matrilineally,[12] although by the time of the Shang dynasty (1600 to 1046 BCE) they had become patrilineal.[12][13] Chinese women do not change their names upon marriage. They can be referred to either as their full birth names or as their husband's surname plus the word for wife. In the past, women's given name were often not publicly known and women were referred in official documents by their family name plus the character "Shi" and when married by their husband's surname, their birth surname, and the character "Shi".
So yeah, given 4000 years of using surnames, why would it be abandoned in another 1300?

(Wow - people will botch about anything to demean this show.)