That’s it. It’s like an Adventuring Party in other tabletop RPGs, there’s fluff around what they are that makes them sound more complicated than they mechanically are…
Circle: A Circle is a group of Exalted brought together by oaths, pacts and sorcery
Exalted 3E p.25
But all other references to ‘Circle’ in the Third Edition book, where it isn’t talking about Circles of Demons or Circles of Sorcery (which denotes hierarchy/power), only refer to the Circle as a group of Exalted. There are no special mechanics for how you form one, nor are there any benefits beyond cooperation. There are no sorceries that create a Circle, nor are there any specific Oaths or Pacts laid out or making one.
It’s not even specific to Solars. It talks about Circles of Abyssals in the Core Rulebook, Lunars are spoken of in Circles in Fangs at the Gate. This was likewise true back in 2nd Edition. The only difference in 2E was that it drew the distinction of a “Perfect Circle” which was 1 Solar of each Caste. But even then, there was no mystical component to being part of one apart from “Now there are five Solars you have to deal with–and with one of each Caste, they should be a well-rounded group.”
Likewise, 2nd Edition referred to a mixed group of Exalted as a “Mixed Circle” and that was, likewise, just a label.
Put simply “Circle” is to Exalted what “Party” is to Adventurers
The one exception to this are the Dragon Blooded…a feature of their Type is that, as soldiers, they are designed to function as a unit–and thus they have numerous powers that are designed to facilitate teamwork. Central to this is the idea of a Sworn Kinship.
The process of joining a Sworn Kinship is…
The oath of kinship can take many forms, but always involves a recitation of the names of those forming it, a statement of intent, and a vow of dedication, spoken as the participants’ animas flare. When the ritual is complete, the newly sworn companions know it deep in their bones.
But it is important to note that this only works for Terrestrials.
The oath of kinship is mystical, a bond born from Terrestrial blood and elemental Essence. It’s a sacred birthright of the Dragon-Blooded, and the greatest boon the Dragons bestowed upon their children, that they might fight as one.
Both quotes from Dragon-blooded, What Fire Has Wrought p. 105
As you can further read on that page, Sworn Kin can sense each other’s presence–and a number of Terrestrial Charms work better or more efficiently when used on or with your Sworn Kin (for example: Virtuous Negation Defense costs half the motes if used to defend a Kin).