Greeks and Romans did not have a prophet who will dictate their dogmas
The Greek and Roman religions have no prophets, no dogmatic books, and have no hierarchy priestly class.
Almost all religions have a holy book, a dogma and doctrine generally transmitted by a prophet or messenger of God. Christianity with his Bible and Jesus, Judaism with their holy books and prophets, and Islam with his Koran and Mohammed are known precisely as "religions of the book". In such cases the words inside the book are the word of God, and therefore its contents are dogmas that can not be questioned.
In Ancient Greece and Rome there is not a prophet or theologian who has communicated a similar doctrine, religious rules of conduct. And therefore religion is a field in which you can act much more freely. There is no holy book like the Bible or the Koran, at most there are books of prophecies, called "Sibylline Books" that the Romans, very superstitious people, consulted in any circumstance of particular importance to the republic, for the State.
Nor is there a group or priestly caste monopolizing the relationship with the divine, though some families may have a special relationship with a particular deity.
The ones who set many of the different characteristics of the Greek gods were Homer, talking about them in his Iliad and Odyssey in the eighth century B.C., and then Hesiod in his Theogony briefly after. They set their names (Zeus, Poseidon, Aphrodite, Apollo, etc..), their figures, their skills and arts.
It should be distinguish the gods whom the Greeks worship in their temples and their cities from the gods of mythology, created mainly by poets precisely thanks to a religion without prophets and without dogmas. And even more, we must differentiate "the gods of the philosophers", the result of reflection on religion that produces all kinds of thought towards religion, from atheism to deism or animism...