What a state faith is?
To me a state faith is the constitutional arrangement and basis for the official recognition of a temple as the primary spiritual source for the realm. This should mean a series of obligations, requirements and responsibilities on both parties to allow the realm to prosper and the spiritual needs of its people to be met. Normally this is done by a contractual arrangement which is acknowledged by the Court of the Realm.
Normally included in such an agreement will be blessings, monetary compensation, hardiness, support and so forth in order to allow for the smooth running of the ship of state.
I would argue that the basis of a state faith is what the temple in question can get agreed. Changing a long established state faith though is going to cause significant stability issues I would say, as well as the ire of the followers of that faith....