To begin, "religion" is a modern and mostly Western category. Cultures that have not been Christianized generally don't have the same sense of religion as something discrete from everyday life. The ancient Greeks and Romans recognized that other cultures had their own ancestral practices, and while they considered their own practices superior, they weren't interested in going around converting people separately from political expansion. This is why the Jerusalem temple was (mostly) exempt from imperial cult--Judean ancestral practice forbade it, and the Romans respected the Judean culture as ancient.

In Greek and Roman societies, what we would categorize as "religion" was just part of daily life--it wasn't separated or considered private in the way that we do it in the modern West. Marketplaces were full of shrines, guild meetings often included sacrifices, the closest thing to a hospital was the Asclepius temple, etc. There weren't major conflicts of practice because changing the way you did religion would entail changing the way you did everything, often in ways that put you outside of society (how do you avoid the marketplace?), and most people weren't interested in that, with a few major exceptions like Pythagoreanism and Christianity. There weren't conflicts of belief because belief wasn't a particular value. Nobody really cared what you thought about Apollo as long as you showed up and did the sacrifices correctly.

If you left Rome and went to Syria or Gaul, you might think that they weren't doing things "right," but for the most part you would recognize their ancestral practices. Cultural influence was certainly a major force, but our evidence for that tends to be Greek or Roman elites moving in and instituting their own practices without major efforts to halt most local practices (at least in the east--I'm less familiar with Gaul and Britain), and non-elites adopting and adapting Greek or Roman practices separately from official civic practices.

There also aren't "factions" of different gods in Greek and Roman culture. You went to the Apollo temple on one day and the Demeter temple on a different day, and that was all fine. Certain gods had more or less importance in different regions and cities, or for different groups of people, but they weren't in competition with each other. The closest you really get to factionalism, besides ethnic differences and philosophical schools, is mystery initiation, but even that usually wasn't exclusive. Everybody in Athens who could afford it (and a whole lot of people from other places) was initiated into the rites of Demeter at Eleusis. There are some accounts of initiation into the mysteries of Isis that suggest it was considered the most important cultic practice for some people (especially Metamorphoses by Apuleius), but they still didn't stop participating in other ones.