The civilopedia explains the Firaxis reasoning for making all religions identical. You can mod religions to have wildly varying characteristics. The main thing religions do is NOT make people happy it is make people Rich. Shrines are a powerhouse. I think that is what should change, though its a good game mechanic corporations can take over from it. Having the shrine doesn't really make your secular government rich directly, it earns you prestige, so that other civilizations with the same religion have a better attitude toward you, depending on how prevalent it is.
Also the selection of religions. Not to debate what is and is not a religion, but religions which have been important historically should be included, especially if there are civs and or leaderheads for it. To have Cyrus of Persia and no Zoroastrianism? The fact that it died out should be irrelevant, this is about history. Pagan type religions while explained away, really have been important internationally and functioned like civ religions. The Mesoamerican, Hellenistic, and Egyptian religions in particular. They had centers of power, shrines, temples, cathedrals, and were widespread over more than one nation. Shinto, important as it was for Japan, never spread internationally. Taoism pretty much was a Chinese belief system, and though Confucianism was international, it lacks many characteristics of a religion, not only in its doctrines but in the way it doesn't really have a structure like a religion.
Perhaps there could be something like Lesser Religions. These require a tech only one nation can have that cannot be traded, so that nation founds them on the game starting. However, they have no shrine.