The difference is that the DCU has, for a very long time (implicitly at times, explicitly at others), embraced and supported the idea that philosophy/theology issues are subjectively true/false. If it's true for you, then it's real. It often gets lazily cross-applied to magic, with magic being categorized as philosophical itself. But it's a real-world idea that DC has used effectively to sidestep questions like this: for example, the ancient Greek/Roman gods are less powerful because fewer people believe in them, etc. It's even been explicitly supported at times that different creation myths, different pantheons, and different general mythological structures are "true" for an individual if the individual believes in them.
This is true between the pagan pantheons, yes, and it may be what they
say is the case for all faiths, but the fact in practice is that the Abrahamic monotheist God has almost always been placed on a higher level of regard and credulence in the DCU, even from other pantheons and magic-users. In Vertigo, the very first thing Tim Hunter sees at the beginning of time is the war in Heaven. Not Olympus, not Asgard, but the war in Heaven between angels. And the current Reign in Hell is not happening in Hades, or in King Yama's realm, or in Hel, but in the Hell of the Bible filled with Biblical demons quoting Bible passages.
And the idea that the DCU gods literally gain power from belief is a bit misleading. That is the case on the surface, but the real crux isn't whether people still believe in them, it's whether they -- and their sphere of influence -- can stay relevant to people. No one cares who Ares is, but war is the biggest deal on the planet, and so he's became the single most powerful Olympian, followed much later by Athena and Aphrodite. The Japanese gods in Vertigo are no more worshipped today than any other old religion but have managed to evolve and change with the times and have therefore retained a lot of their power.
The Swamp Thing story that was supposed to establish that got ****canned. You and I can say that's the case as much as we like in our personal continuities, but I don't think it's established in canon.
I'm talking about the Ostrander Spectre which stated that Jesus was God's spirit of mercy on Earth, and the Spectre was in limbo while Jesus lived because the spirit of wrath and mercy couldn't coexist at the same time, and that ever since then the Spectre had to be bound to a human host to function.
Why would that necessarily be true, though? It's not like the Guardians have the Corps set up in other universes. They're clearly only a presence in the mainline Earth. So there's really no reason that there can't be Guardians for other Universes.
Don't ask me, ask whoever designed the multiverse back then. *
shrug* The rule is that there was and should always be only one Oa. It has something to do with the fact that an Oan, Krona, was the one who originally looked back in time and shattered the universe into a multiverse in the first place.