Word for the religious equivalent of 'demonym'

Since demonym derives from "demos," and refers to a village, or more in more general modern usage, a place (nation, continent, etc.) perhaps "theonym" might be an appropriate term to coin (if, in fact, this has not already been coined...). It's not exact, since "Theos" translates to "God," and so doesn't literally indicate a categorization by religion, but by deity I suppose, I think it appropriately gets the point across.

Is ethnonym acceptible?

"is a name applied to a given ethnic group."

https://en.wikipedia.org/wiki/Ethnonym

https://www.encyclopedia.com/philosophy-and-religion/christianity/protestant-denominations/shakers

https://www.encyclopedia.com/philosophy-and-religion/judaism/judaism/jews

Being as a religious grouping is a cultural grouping, ethnonym looks appropriate, indeed given your 'muslim' example it does appear to be correct.

'Ethnonym' can also apparently be used to describe lower order terminology.