We don't know a huge amount about how this worked, but it was probably based on their region and their particular customs. These religions are "polytheistic" precisely because there was no single book or doctrine that people agreed on. They did not have the equivalent of a "Bible" to explain these things.

In Greece, each community would have one or more temples with a cult dedicated to a particular god. But they didn't necessarily have temples or cults for every single god, and the teachings of one cult probably didn't exactly match the teachings of another cult in a different city, even if they both worshipped the same god.

I suspect it was a judgment call based on factors like whose temple was nearby, who was considered the patron of a certain city, and which god's offerings had seemed to be successful in the past.

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Person A: "My father told me the story of how he sacrificed a goat to Athena, and he lost that battle. So maybe we should try Ares."

Person B: "I don't think so. The other guy's General claims he is descended from a son of Ares."

Person A: "Okay, so we stick with Athena but we try a cow instead of a goat?"

Person B: "Yeah, that sounds good."