memes

The reality here but quite a bit more complicated than I think is commonly understood. One thing to remember is that both of these groups (the Greeks and the Italics) actually shared a common religious source. They were both indo-european and already shared a common base.

There is also the fact that the Greeks established huge numbers of colonies in the Mediterranean. In fact, the entire southern bit of Italy was known as Magna Graecia. Greek culture was sort of ubiquitous in the Mediterranean. In fact, some of the Greek stories that made their way into Roman religion, iirc, came through Etruscan borrowings from Greek religion, which the Latins then borrowed from the Etruscans.

If you want to go even deeper, look into how much the Greeks took from the Egyptians and the Anatolian "Hittites". It turns out that stealing gods and stories was (and is) a common theme amongst people groups. The Greek goddess Aphrodite was probably (emphasis on probably) a loan and reinterpretation of Ishtar/Asherah/Innana. Look up Serapis as well. Fascinating examples of syncretism by the Greeks.

This was not new by any stretch of the imagination, nor was it uniquely Roman. We can talk about syncretism in the middle east and the Indian subcontinent next if you want, lol.