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We know that Melisandre knows the shadowbinder techniques. We've got no idea as to who the ordinary people of Asshai worship, so I'd hold off making shadowbinders and red priests the same. Melisandre probably could have the shadowbinder trait though as a way to represent her skills though.
 
It's more like the case of Qarth. We don't know which religion is exactly the one popular there ( probably there's no single one, since it's a mercantile port), so I guess the AGOT team took the closest pseudo religious group the could find (the Warlocks) and made them into Qarth's religion.

My guess for it's "state" religion though, is that the Pureborn represent some kind caste worshipped in a Imperial cult fashion (like the Roman Emperors claimed to be "Divi filius" or the various Egyptian rulers were considered living gods by the population). In ACOK it's mentioned that a sacrifice must be made at a temple in order to receive an audience with the Pureborn. IIRC, it doesn't mention to whom the sacrifice is offered to, but we can assume it's to the Pureborn and not to the Warlocks... who instead are seen as pariahs who hide in a tower practising pointless "magic" rites (until well... Daenerys happens) and are only paid lip service.

The case of Asshai is even worse, we're given little to no information of it, the actual city of Asshai I mean, not the Shadowlands and whatnot. Again, same as Qarth, the only pseudo-religious organization we know of originating there are the Shadowbinders who've trained at least 3 different characters in the series, though all of them either profess another creed (Mel and Mirri Maz Duur) or are undefined (Quaithe), it's the closest there's to a religion in Asshai. After all, a massive following of R'hllor wouldn't make much sense in the "Shadowlands", and there doesn't appear to be a Red Temple over there, they're mentioned to be all over the world (Oldtown and Dorne even), but pointedly not Asshai. There's only "ancient books" of prophecy about Azor Ahai there, presumably a part of a larger library of arcane knowledge.
 
shadowbinding is a type of magic, like greenseeing, blood magic etc. Rullor is a religion. They are not the same, they overlap.

The only R'hllor follower we have ever seen use Shadowbinding is Melisandre and GRRM himself has confirmed that she is a renegade with her own agenda and not a representative of the greater Red Faith.
Melisnadre also is not from Asshai as in her chapter she thinks "that was a lesson that Melisandre had learned even before Asshai". Melisandre went to Asshai to learn Shadowbinding, just Mirri Maz Dur did, because it is something unique to the peoples of Asshai.

Quaithe is the only Asshai'i character we have actually met and she is a Shadowbinder with no connections to R'hllor. At the docks in Qarth, Dany sees boat loads of "masked Shadowbinders out of Asshai". So wether or not it is a true religion, it is something unique to Asshai and so in the absence of any better information serves the purpose adequately.

As you say it is a type of magic similar to Greenseeing, and I say to that, if the Old Gods religion wasn't named in canon, in the mod it would probably be called "Greenseeing".