The only game with a Religious Victory Condition is Civ6, as
@Alexander's Hetaroi wrote. Religion was implemented as a customizable part of one's empire, with its own yield as in Civ5. The faith yield was better integrated with other mechanics, allowing purchase of civilian units, military units, and buildings with faith when the proper requirements are met. All of those aspects interacted with the player's ability to pursue the other VCs.
Religion in Civ5 was implemented as a customizable part of one's empire, with its own yield (Faith). Religion could influence Tourism (BNW), which is important for the Cultural VC.
Religion in Civ4 had a huge influence on diplomatic relations. A fixed number of religions were defined in the game, unlocked with different techs. Adopting a state religion -- or not, depending on your civics choice -- influenced your economy. If a religion was dominant in a city, one could build buildings for that religion, earning culture, as well as train religious units. The cultural VC in Civ4 (BTS) was heavily dependent on having religious/cultural buildings in certain cities. I could make a case that the late game corporations in Civ4 BTS had a similar/parallel function to religion: building branches in a city, spreading the corp by training executives, yields in the city influenced by the corp branch in that city.
Religion didn't exist in Civ3. Certain buildings (Temple, Cathedral) or wonders with religious-sounding names provided culture and happiness.... essentially, culture buildings.