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.