Words ending "nym" are words that describe types of words. Examples of demonyms are "Briton" or "American". "Demonym" is a recent coinage. It was invented in the 1980s. If you want a word that describes words like "Christian" or "Muslim" you are out of luck. No single word exists.
While we often talk about religions, we don't often talk about the words that name religions, so no single word has been developed for this category of words. Just describe:
We don't need to say "The xxxonym of Islam is Muslim" because we can say:
The word used for a follower of Islam is Muslim.
"Christian", "Muslim" and "Bahá’í" are the words used for followers of Christianity, Islam and the Bahá’í faith, respectively.
Or, for example:
Adherents of the Bahá’í Faith prefer to be known as Bahá’ís not Bahá’ísts.