Nobody knows. The source material just isn't there. We can't even be sure of Cyrus' ethnicity.

The inscriptions of Darius I and Xerxes I (Behistun and XPh, especially), which are heavily loaded with religious language, makes it clear that they were devout Mazda-worshippers (i.e., Zoroastrians) who were intent on enforcing Mazda-worship on the Iranian population. Darius I appears to have extended this to the Elamite population, demanding that they worship Auramazda (the OP spelling of Ahura Mazda), in exchange for sponsoring temples to their native deities (per Persepolis temple tablets).

But as you probably know, the connection Darius purports between himself and Cyrus, common descent from an alleged King Achaemenes and his son Teispes, is probably spurious, given that Cyrus, in the Cyrus cylinder, only traces his family tree King Teispes of Anshan. Moreover, the names of Cyrus' family members, except for his daughters (whose names, as far as I know, are only attested in Greek sources), are Elamite, including Cyrus. We can see this for the name Cyrus because in the parallel inscriptions at Behistun, in Akkadian and Elamite, the name is written Kuraš rather than, as you would expect if it were a Persian name, Kuruš. The Elamite form is attested elsewhere and has a clear etymology (Caregiver) in the Elamite language.

My interpretation of Darius' claim is that Achaemenes (a Persian name) was just the name of the eponymous, possibly semi-mythical, ancestor of his clan, the Achaemenids, otherwise it would not have made much sense to insert an additional paternal ancestor after Teispes. I wish we could understand what was going on in Darius' head, but I'm afraid that's beyond the reach of history.

So for Cyrus' religious beliefs, we basically have the following:

• Probably an Elamite, king of the Elamite city of Anshan, presumably a worshipper of Elamite deities.

• Has no problem with dedicating his reign to Babylonian deities in the Cyrus cylinder, but it is an extremely formulaic document which we can't attach too much weight to.

• Elamites had a significant cultural overlap with Persians, and it is not impossible that Cyrus worshipped Auramazda, given that Darius saw it fit to compel other Elamites to do so. On the other hand, Darius and Xerxes were essentially theocrats, and could just have seen this as an appropriate thing to do to shore up their claim.

• Cyrus was a semi-mythical figure already during or shortly after his lifetime, so I attach little to no value to the stories of the details of his conquests and famed tolerance attested in Greek and Hebrew sources.

That, as you can see, is pretty much a long-winded way to restate what I said at the outset: Nobody knows, and we probably never will.