The claims vary — either that their religion was purely ritualistic, but they didn't take it literally, or it was not mystical, or some such nonsense. The case seems unique to Greece and Rome, and no one ever claims that say, Egyptians or Babylonians were the same way.
What I'm wondering is if it's, at least on some level, the legacy of the Renaissance — could it be said that the antiquity got somewhat 'secularized' by Italian humanists to present the ancient authors in a way that would better fit contemporary times?
Well, it actually goes back to the Greeks. A number of Greek philosophers of the fourth century express skepticism about the traditional religion. Xenophanes famously mocked the anthropomorphic depictions of the gods, saying that if horses and cattle and lions could paint, they would paint their gods as horses and cattle and lions and pointing out that other peoples depict their gods so that they look like themselves.
Xenophanes is only a small part of a greater intellectual trend, one that can be seen in the negative depictions of the gods in (particularly Euripidean) tragedy, the mockery of gods in Aristophanic comedy, or the outright criticism of myth and beliefs in Plato.
Of course, this was - to some degree, at least - an intellectual pretension and probably not a genuine reflection of the life of the "average" person. Also, the survival of this type of argument was propagated in late antiquity by the early church fathers. In fact, it is probably a significant reason why some pagan texts escaped zealous early Christians, because there was already a precedent for reading them as "just stories" rather than as religion.