It's a bit murky. As far as I know, it's not explicitly addressed in the film or written materials - beyond that, in the novelisations, apparently in descriptions he's referred to simply as "Ren" (as in, "Ren leapt to his feet and raced to the door, leaving his eyeliner and poetry unfinished"). If they all end their names with Ren like it's short for "[the] Kylo [of the] Ren", that could get confusing quickly.
This might suggest "Ren" is the unique part and "Kylo" the part everyone shares (like "Sir Ren"). The "Ren" part would then either be his adopted name (which gives us the question of why it's also the name for the order of Knights of Ren - easy to answer if our Ben founded the Knights, or joined them by subverting them into their current form, difficult if he joined as a regular member) or the title given to whoever the current leader is, like "Dalai Lama" in real life (making "Kylo Ren" effectively "Sir Boss, boss of the Knights of Boss").
Or this might be an over-interpretation of a fairly ordinary writing convention. There's not enough evidence yet to be sure.
The closest I'm aware of to an official comment is this quote from an Entertainment Weekly write-up of an interview with Abrams:
Remember how we eventually learned that “Darth” is not a first name, but a kind of title? It appears the surname “Ren” is something similar.
“He is a character who came to the name Kylo Ren when he joined a group called the Knights of Ren,” Abrams says. But that’s as far as the writer-director will go.
(Note: this interview was before film came out- I'm not aware of anything more recent on this)
This has been interpreted as a sign that other Knights will have names like "Brian Ren" and not names like "Kylo Stimpy", for example in this quote from cinema blend which I found in this answer to a similar question about whether his name is based on other characters' names:
As we learned in an interview with Abrams in Entertainment Weekly, "Ren" is the title, which refers to a mysterious organization called The Knights of Ren. "Kylo" is the unique signifier, a name Kylo Ren chose for himself for some equally mysterious reason
But they might be over-interpreting EW's over-interpretation of a vague statement. Unhelpfully, Cinema Blend don't link to or reference which EW interview or write-up they're referring to, but I can't find one that comes any closer to a confirmation than the one I quoted from.
So that's two weak clues that appear to point in opposite directions...
The overall answer seems to be, we just don't know yet (unless there's an interview or official comment I'm not aware of). They appear to be being deliberately secretive about the Knights of Ren - presumably they'll play a major part in the sequel.