Make the Church spawn "religion students" which are virtually useless, and not buidable by anone.
Then make it upgrade to a chain of Shaman, Priests, Bishops, etc. but make these units kings, and civ specific. It won't be possible to build them directly as they are king units, but you can upgrade the "students" to them, and each civ can only upgrade it to the religious units it is allowed to have.
And this way, with only one building, you have as many civ specific religious units as you want, spawned by building