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.