Association between antagonists in Iron Man 1 and Iron Man 3

Since the OP said that he is open to answers outside the movie universe here goes this:

The Mandarin is portrayed as a genius scientist and a superhumanly skilled martial artist. However, his primary sources of power are ten power rings that he adapted from the alien technology of a crashed space ship. Each ring has a different power and is worn on a specific finger. From Wikipedia

So, the Ten Rings terrorist group in Iron Man I, was a reference to that:

The Ten Rings was a shout out to Iron Man’s archenemy in Marvel Comics, the Mandarin, and refers to the literal ten rings that the Mandarin wears, rings with alien origins. From screenrant

Now the short film All Hail the King, says that the real Mandarin exists:

Ben Kingsley reprises his role as Slattery in the Marvel One-Shot short All Hail the King. In this film, Slattery is an inmate at Seagate Prison being interviewed by Jack Norriss, a filmmaker documenting his life. However, in a plot twist, it is revealed that Norriss is a member of the real Ten Rings organization led by the real Mandarin, and that they are angered by Killian and Slattery making a mockery of their beliefs. He kidnaps Slattery from the prison, telling him that the Mandarin wants to meet him. From Wikipedia

I don't know if they plan the Mandarin to have those literal ten rings of power from another planet, or only the organization called the ten rings as a reference to the comics.

With Capitan America, Thor and Avengers introducing super natural powers from other planets and universes to the movie universe, I think they could introduce the real Mandarin with the rings as it is in the comics without been unrealistic in this continuity.

There certainly is a relation between the two. While one might consider the mention of the Ten Rings in Iron Man 1 as an easter egg for the fans, it seems that the filmmakers have always conceded that the Ten Rings and the real Mandarin exist in this onscreen universe.

SPOILERS BELOW

Indeed, in the new short film that can be found on the Thor 2 DVD, we learn that the Mandarin does exist and is displeased with Trevor's take on him.

From the short film All Hail the King:

Jackson Norris: I'm not the one that's going to kill you Mr. Slattery. I'm here to break you out.

Trevor Slattery: That's... nice?

Jackson Norris: Not really. You see, there's somebody who wants to meet you.

Trevor Slattery: Do I know him?

Jackson Norris: No... but you took his name... and now he wants it back.

As for the relation between the Mandarin and the Ten Rings, this excerpt from the Marvel wiki explains a bit about their mutual history.

Discovered by the Mandarin in China’s “Valley of Spirits,” the Ten Rings of Power are actually the product of the dragon-like race Axonn-Karr (or Makluans) from the planet Maklu-IV. The Axonn-Karr utilized them mainly as the power source for their interstellar craft’s engines. The rings’ functions currently cannot be explained by modern Earth science.

The Mandarin learned how to utilize the rings for own personal use and make them respond to his personal commands.

and then

The Ten Rings is the name of the terrorist organization that abducts Tony Stark in Iron Man. The organization's name and iconography was later appropriated by Aldrich Killian who falsely claimed that he was their leader, The Mandarin, and placed blame on the group for his illegal activities.