This is true, but it was never precisely practised as such. Religious practice, and the stories lying in the background of that religious practice, are really separate things for most purposes: there are links between them of course, but the nature of those links will depend on a variety of things.

In the case of Greek myth: in antiquity, the link between religious practice and myths usually lay in a specific story having a significance for a specific shrine. The cult of Demeter and Persephone at Eleusis, for example, was the religious frame for the story of Hades' kidnapping of Persephone; the cult of Apollo on Delos was the frame for the story of Apollo's birth there; the cult of Hades at Elis was the frame for the story of Hades coming to the aid of the Elians when Herakls sacked the city of Oichalia; the cult of Odysseus' wife Penelope at Mantinea was the frame for the story of Penelope giving in to the suitors and having sex with all of them, then giving birth to the god Pan ("all"). And so on.

There were cases where a specific shrine incorporated multiple stories into its temple decorations and iconography: the Parthenon temple in Athens includes the story of the battle of the Lapiths and Centaurs in its decorations, a group gathering of all twelve Olympian gods, and other scenes. When that kind of thing happens, it's really intended to expand the importance of the cult centred on that shrine by laying claim to extra myths.

And of course there are many myths that don't have any known tie to any specific cult site. In those cases, those are myths where we don't have good data on any particular religious practice linked to the myths.

So as I said, even in antiquity it makes most sense to talk about mythology and religion as separate things. Religious practice was tied to specific cults at specific shrines in specific places. Mythology, by contrast, was something more fluid that sometimes had a tie to a specific shrine, but was to some extent the cultural heritage of all Greeks.