Despite being a new addition, the Ten Rings are already apparently famous in the Marvel Universe, because at least one dignitary knows about them and their insignia. That insignia is a key detail linking the Ten Rings to the Iron Man movies, and we get a good look at it towards the end of the issue, when Clayton low-key hands one to Riri. The insignia also comes with a rather cliched, "you and I are the same," but that's probably why Clayton is a C-tier Spider-Man villain and not the next Mysterio.
The insignia Clayton passes to Riri bears a striking resemblance to the flag seen in the first Iron Man, which hangs behind a group of terrorists as they read the list of demands that must be met if Tony Stark is to be ransomed. There are, of course, some distinctions. Namely, the symbols inside each ring. The difference might have a thing or two to do with the history of the Marvel Universe Mandarin's ten rings. Not a terrorist group called the Ten Rings, mind you, but ten actual rings, originally worn by the Mandarin. Oh, and they're from space.
See, while the Ten Rings terrorist group doesn't exist in the Marvel Universe, the group is based off ten powerful, cosmic rings worn by their leader in the comics, the Mandarin. Introduced in Tales of Suspense #50 in 1964, the rings themselves aren't even really rings, they're just powerful cylinders (crafted by an alien race named the Makluan) that look like rings to the fabulously ignorant eyes of a supervillain. The rings offer their wielder devastating powers, like the ability to produce poisonous gas at will or cloak an area in darkness, and, of course, shoot all manner of lasers.
Here's where things get interesting, though. Eventually, the rings were activated by an alien signal and scattered across the world, latching onto various villains and everyday people. Those ten people were never referred to as the Ten Rings (if anything, they were called the Mandarins), and they didn’t even really work together. The Spectral Ring ended up going to the Dark Elf Malekith, who made it a sport of trying to kill the other ringbearers, so he could collect all ten for himself.
It's possible the rings that empowered the Mandarin have returned and somehow latched onto new hosts. It's also possible these new hosts have assembled to create a new terrorist group that people like Clayton are jumping to join. Which brings us back to the different symbols on the comic book version of the Ten Rings insignia, each of which could correspond to an individual ring and its power set. It's worth noting that the rings were only taken from their bearers when Tony Stark went all Lord of the Rings and created a Master Ring… you know, to rule them all?
Stark has had a rough time of it lately, having essentially been killed and resurrected within the span of a couple years. Something could have happened to the rings while Stark was presumed dead, again unleashing them upon the world. If that's the case, what happens if someone else gets their hands on the Master Ring? The members of the Ten Rings wouldn't even have to play nice with each other if they're being controlled by the Master Ring.
As for why Marvel and writer Eve Ewing opted to use the name for the company's latest supervillain group. Well, that should be the most obvious part. Ten Rings is just a wicked cool sounding name, and now they've come to the Marvel Universe.