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". If they all end their names with Ren, that could get confusing quickly, which might be a point in favour of "Ren" being some kind of unique name, or title given to the current leader (like "Dalai Lama").
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 I think the overall answer is we don't conclusively 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.