In regular BtS, and any mod I can remember playing, a Shrine doesn't let you train missionaries. Only the monasteries do.
I made a mod that does that a long time ago (and I've been adding and adding to it since then).
Although, it didn't train "missionaries" per se.
It trained "Prophets" (not great Prophets) - which looked and functioned the same as missionaries, but had double the religion spread value (which I think determines its chances of success in spreading the religion), 3 movement points (instead of 2), and cost half the hammers of a normal missionary. It also used a different limit, so you could have 3 missionaries, and 3 "prophets" - making it even nicer to control a holy city with a shrine.
Later, I added an "evangelist" class, that was available after mass media, that required a "revival center" or "religious center" for their respective religion.
The "evangelists" functioned the same as missionaries, and the building required for them was cheaper than a monastery (since it only gave 1 culture, and no research boost).
In this implementation, any city could build them, but it would be easy to make this "religious center" require the shrine.
There are a lot of options with just XML if you don't insist that they be called the same name in the xml file (you can have two different units have the same displayed name in game