Why does Homeric religion look more classical Greek than Mycenaean?

Yes, Hellenistic (4th-3rd century BCE) Greek chronographers liked to date the fall of Troy to the 1300s-1200s BCE. And that happens to line up reasonably well with the end of the historical Bronze Age as we understand it. Troy had been resettled by Greeks in the 8th century BCE, and the contemporary city coloured all of the legendary material that arose around it. Homeric epic in particular is almost purely 8th-7th century BCE: in terms of its depiction of material culture, marriage customs, military equipment, inheritance customs, legal and political framework, and so on.

This naturally goes for religion too. The Iliad depicts the main civic cult of Troy as that of Athena Ilias, ‘Ilian Athena’, the cult established by the 8th century Greek colonists. The Bronze Age cult of Appaliunas still existed, as a cult of Apollo, a few km from the city, but Athena is right there in the middle of the city in Iliad book 6. Similar things apply to the rest of the pantheon.
The story of Homer being ‘memorized for hundreds of years rather than written’ refers to the period between the poems’ composition and their transcription, not the period between the Trojan War and Homer. 20th century scholars had to posit centuries of oral transmission, because in the mid-1900s the composition of the Iliad was conventionally dated extremely early. Finley thought ‘Homeric culture’ was 10th century, for example; but the classical alphabet only emerged in the 700s. The ‘centuries of oral transmission’ were invented to explain that discrepancy. In more recent decades the epics have been gradually but consistently down-dated, for a variety of reasons, especially the references to 7th century material and military culture that I mentioned above. Nowadays the normal dating is ca. 670-650 BCE for the composition of the Iliad, and a slightly wider range for the Odyssey. As a result it’s possible to imagine they were written down straightaway, and some scholars believe precisely that — though as it happens I tend to believe they were transmitted orally after that point, for at least a century, and only transcribed in the late 500s. Either way, no one’s talking about Bronze Age stories being memorised. (Except for a very few, very elderly scholars, who just refuse to accept the down-dating of Homer and Homeric culture.)