I'm not familiar with this kink.

100% of respondents thought you were asking why travelers use host names even among themselves when no one else is around.

Was that your question?

If your question is why travelers used host names among themselves even when no people from the present were around, it is straightforward.

Otherwise, the fact that everyone was answering the above question instead of what you actually wanted to know proves that your question was not straightforward.

Was your whole question really supposed to be about their future names?

It looks like everybody understandably took your main question to be what you actually started with, "Why do the travelers keep the names of their host bodies?" plus "I understand when interacting with people from the present. Why keep the ruse amongst themselves?"

The part about future names seemed easy to overlook if you're assuming that all of the questions were supposed to be ways to elaborate on a single topic of the value of sticking only to the host name in all situations.

Forgetting all that and just addressing future names, in season 3, episode 9, "David asks Mac what his real name is, in the future. Mac gives Marcy the side eye for letting David know about the future- maybe there’s still time to give him some memory serum before he dies. But he tells David his name is 3468" (quote from a recap).

There's no reason for Mac to lie, though logically we can't rule it out. Barring any further clues, it does seem like Mac's only future name is this number. I'm not aware of there being any official explanation, but if I had to write the backstory on that, I'd guess travelers would probably be required to abandon their original lives and identities in service of the program because they needed to be ready to become the host identity as much as necessary without distraction or conflict created by clinging to any past names, ties, or habits. Assigning a number instead of a name would be one way of maintaining clear identification without reminding them of their original lives.

That's the point.

If they have to remember to call each by their host name in public and traveler numbers in private, that means that they get a lot of time practicing using traveler numbers out loud to each other.

The more you practice something, the easier it is to do. Because of that frequent practice of using the traveler number, they would become more likely to accidentally use the traveler number in public and raise questions.

If instead they use the traveler number as little as possible, and constantly practice addressing each other by the host names, it reduces the possibility of accidentally using the traveler number in front of non-travelers.

They live their full lives in that host, including times when they might be drunk, injured, or otherwise mentally not at their best, and in any state under any circumstances they must continue to maintain the illusion that they are the hosts or they will risk the success of the entire program (consistent with what Spartyjason said about going undercover in general).