Yes, of course Star Trek virtually always assumes that every alien race has only one name, one language, one culture, one religion, one hairstyle, one fashion sense. I'm not saying that isn't the case on the show, I'm saying it's stupid as hell that it's the case, because we're talking about entire species that must have thousands of years of history just like humans do. Logically, no matter what lazy assumptions the writers make, any alien species should have hundreds or thousands of different current and historical languages and many different names for themselves.
IMHO, in fiction, stuff like this is a reasonable sacrifice of realism for clarity's sake.
Also, there have been shown to be some variations within cultures, like the Klingons. Case in point, in "Looking for Par'mach in All the Wrong Places" [DS9] (or however that word is spelled), Worf makes reference to the idea that a Klingon lady comes from a specific region of Q'onoS with its specific traditions. A lot of TOS Klingons have shorter hair than in other shows (maybe simply a fashion of the times, but still). Some Klingons believe that any form of capture requires suicide to remove the disgrace ("Birthright, Parts I and II" [TNG]), while other Klingons simply live in disgrace after release ("A Matter of Honor" [TNG]), or even think that as long as there's a chance of escape, they haven't been dishonored to the point of suicide ("In Purgatory's Shadow" [DS9]).
That's not even going into the differences we see in Vulcans, from ENT to TOS onwards, Bajorans (some religious, others agnostic), the more death before dishonor Romulans of TOS/TAS and the slyer ones in TNG onwards, etc. So, while I will concede that
Star Trek does play the monoculture card a lot, they also add some diversity to the ones that get more exposure (the universal translator also makes it hard to determine actual language usage).
So the name used by other cultures is going to be whatever name they pick up from the currently dominant culture -- for instance, other species call us "human" and our planet "Earth" because the dominant culture in Trek humanity is an English-speaking one. But that doesn't mean that English is the only language humans have.
While English seems to have become the dominant human language in the future, other languages do continue to exist ("The Lorelei Signal" [TAS], "11001001" [TNG]) (I've always wondered if when Picard visits his family in "Family" [TNG] if they were actually speaking English, or if it was French "translated" into English for our benefit.)
Also worth noting that there have been some alternate names for humans; "Earthmen" on the Romulan side of things, and "Earthers" for the Klingons.