Subjects Temporal and Spiritual

English does not provide a word for those subject to spiritual authority that is any way distinct from the word used for those subject to temporal authority (meaning secular or civil authority). We have always used subject for both of those.

Its main etymon from classical Latin per the OED is:

classical Latin subiectus (masculine) person ruled by a monarch or sovereign state, subordinate, dependant, in post-classical Latin also person under the spiritual guidance of a priest or pastor (6th cent.), vassal (10th cent.)

(There is also a post-classical Latin etymon subiectum (neuter) used for a grammatical, musical, or legal subject.)

They show these relevant senses:

I. Someone or something under a person’s rule or control.

1. a. A person who is under the control of another or who owes obedience to another. Also in extended use of an animal, etc. Frequently with possessive or of-phrase. Also with †to.
2. a. A person bound to another by an obligation of allegiance, service, or tribute; spec. a feudal tenant or vassal; (hence) †a dependant, a subordinate, a person of inferior status. Now historical.
   b. A person who owes allegiance or obedience to a spiritual superior. Now chiefly historical.
   †c. A person under the spiritual guidance of a priest or pastor; any of a parish priest’s parishioners. Obsolete.
3. A person owing allegiance to and under the protection of a monarch or government; a person (other than the monarch) living under a monarchy.

So our English word came to us already possessed of senses both temporal and spiritual when we initially grafted it into our language from Middle French by way of Anglo-Norman French. Indeed, I know of no other word used for the distinction you seek in any of the related modern Romance languages, including French (sujet), Italian¹ (soggetto), Spanish (sujeto), or Portuguese (sujeito).

If by chance you happen know of one commonly used now or in the past in any of those Romance languages, I would quite be interested in learning it — preferably without having to first read through reams and reams of Old Castilian written a millennium ago about the Caliphate of Córdoba that existed there under the Umayyad dynasty during 929–1031. :)

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Footnotes

1. As pointed out in comments, for people who are subjects of a ruler, Italian uses a slightly different word, suddito. I mentioned Italian soggetto because it is equally descended from the same Latin etymon subiectus/subiectum < subicio/subicere/subieci/subiectum < sub + iaceo/iacere that the others use, including English via French.
   
   But Italian uses soggetto only for the “topic or theme” senses of subject — like those from the grammatical, legal, and musical domains. For people like the ones the asker wanted to know about, ones who (are subject / have submitted) to some authority, Italians instead use suddito < Latin subditus < subdo/subdere/subdidi/subditum < sub + do/dare.