Greek myth and religion are two different concepts that are highly intertwined but nonetheless separate, I think you need to be a little more specific if you want a proper answer. I think what you're getting is that the Romans and Greeks shared a pantheon of Gods, and you want to know if the Romans and Greeks were aware of this fact.

I'm a layman that's taken a few classes and read a few books, but the short answer is yes. "Greece" as we think of it today wasn't quite the Greece of antiquity, instead you could think of the Greek diaspora across the Mediterranean as an extension of Greek culture and identity. One of those extensions was Graecia Magna, which pretty much encompassed all of Southern Italy. From the earliest inception of Rome they were bordered by Greek settlements. Even the original demographic makeup of Rome might have included a number of Greeks. Inevitably all that Greek culture was assimilated into Roman life, and for a time the Romans thought very highly of Greece and its culture.

The Romans actively tried to assimilate themselves into Greek myth & religion. A revised founding myth of Rome was that Romulus was the son of Aeneas, a Trojan who escaped the sack of Troy. You may think that this is point of differentiation with Greece but it's more of a homage to the Greeks because Trojans because were depicted as honourable, proud, and fairly Greek-like in the Illiad. It's almost as if to say that the Romans were there with the Greeks all along, inserting themselves in Greece's mythological history. Richard Miles, in Carthage Must be Destroyed, speaks about how Hercules was later said to have traveled through Italy during his trials and adventures. This was partly a ploy by the Greek cities of Graecia Magna to legitimize themselves as true Greek cities, but it was readily adopted by the Romans.

So, yes, the Romans knew that they shared Gods with the Greeks, and it's fair to say the Greeks knew that they shared Gods with the Romans. I can't speak to the Greeks knowing that the Romans had tried to insert themselves in Roman myth, I'm not sure about that. Im sure someone who more expertise in this field can give you a better answer though.