To the extent that it is an organ in these governments, it has a specific organization, with its own officers and doctrines specific to each country. There's no way it could be established otherwise.
I think you are showing an ethnocentric bias. It does not. Any imam can interpret the Koran in their own way - and they often do so. You may have noted the recent comments from Khomeini's son. I'd dare say his views aren't mainstream. But he is still an imam, with a following. There is no "official" church, nor any "official" church doctrine. In fact, there is a fair argument that this kind of "establishment" is far more fraught with peril than a more centralized one.
Anyway - the idea that their are some faiths with fall within "thereof" which do not fall within "religion" is among the most tortured Constitutional constructions I've seen, and is worthy of none other than William O. Douglas in its lack of textual support.