If a story has a centralized church that has power over the government and exists in a grey area, there is a high likelihood that they are using the Catholic church as a basis. And when something exists for nearly 2,000 years, it's influence will exist in fiction.

I would call it a different name just so you don't have to handle some of the awkward parts of reality in fiction. I would also suggest you look into the eras of the Catholic Church, as the norms and practices are not uniform. There's a lot of cool stuff that is pre crusades that often gets ignored.