The Moon In Your Head
Do you really need a physical moon?
Unless your world needs a moon's influence specifically for the tides and similar geophysical phenomena, who says the moon in question has to be a physical object at all? Ancient mind-affecting magic projects the glowing lunar orb onto the vision of the world's inhabitants. Those around the same area will find that they agree on its relative position in the sky so long as they don't measure it too closely, but those far apart will easily find contradictions in its position - perhaps implying false conclusions, such as that the world is a different shape than what it actually is.
In order to create the phenomena of a physical full moon's light, their vision could also be made slightly more effective at night, or light could be cast by an atmospheric phenomena. (Countless bioluminescent organisms photosynthesizing in the upper atmosphere, and releasing their stored energy during the night? And, at that altitude, there's no requirement that those organisms are small, either.)
MindMoon can exist alongside a physical moon
If a physical moon IS required, someone could have sewn particularly powerful invisibility magic around it, or the same mind-affecting magic could make the true moon functionally invisible to the world's inhabitants. Their eyes can physically see it, but their minds are prevented from recognizing it. Perhaps by using physical, camera-like devices, they could find evidence of the physical moon's light and phases, and evidence that the mentally-projected false moon did not actually create physical light (because their nighttime photography is less illuminated than it seems like it should be, compared to taking photos in apparently comparably-dimly-lit areas during the day). For that matter, can other animals see it, or is it only mentally-projected into the minds of humans? Presumably, ancient magic would not prevent humans from recognizing these secondary effects of the physical moon and missing interactions of the nonphysical always-full moon, just directly perceiving the physical moon itself.
"In your head" doesn't have to mean "unreal"
There's also no reason that the mentally-projected moon has to be something unreal; just because the world's inhabitants can only see it through mental projection, and it doesn't have a local physical presence or reality, it could still have a mental, spiritual, or magical reality. It could be a figment woven into the collective psyche of the inhabitants of the planet, where it exists in some shared underlying mindspace. Sight, after all, is a mental process that apparently happens in our brain / mind after our eyes receive the light reflected off of something. In this sense, people who can see it will all agree on the mentally-projected moon's attributes, because they're all seeing the same thing. (Or, the mental projection could simply force everyone to see the same thing.) Is it possible to venture to that nonphysical moon's position through mindspace? Could be.
Why, though?
But why? Why use magic to project a moon into everyone's minds? This is a huge amount of effort to go through, and could even be a fair amount of effort for a god to go through, . I can see several scenarios in which this might make sense:
Something to hide
If an ancient, powerful civilization wielding powerful magic wants to live on the moon and not be bothered, this is a (somewhat) reasonable way to do it. Why make the moon full all the time? This could be to give the humans / other species of your choosing an advantage by only making the mental projection affect them, and grant them better night vision. Adding phases to the moon just makes the whole thing that much more complicated, really, and such powerful magic written into the shared unconscious of an entire planet would be a huge undertaking as it is.
Once complete, the other inhabitants of the planet will notice only that the moon stopped having phases, and that it's always full now, a strict improvement from how it was before in a lot of ways. It's now easier for people to function at night. The constantly-shining moon will be seen as a point of celebration, and perhaps I've read, heard, and watched too much fiction, but myths of a beforetime when things suddenly got moderately better just seem fundamentally less suspicious to me.
Now, what necessitated hiding like this? Perhaps, ya know, ancient Other Man Grew Proud and spacewarred with the ancient Fancy Moon People, and after brutally quashing and exterminating Other Man, the Fancy Moon People decided to hide their existence so that the other civilizations don't come and bother them. Maybe they even feel bad now about raining destruction upon that terrestrial civilization. Maybe they realized that destroying an entire civilization, even in self-defense, is a fundamentally awful act.
Of course, if the terrestrial civilizations find their way to the physical moon again, will they realize that, in the countless years since the Fancy Moon People went into hiding, they may no longer be the same genocidal people who brought catastrophe on the planet? And, does that matter?
Something that we're better off not seeing
I previously mentioned the possibility of seemingly endless swarms of high-altitude bioluminescent organisms which, unlike terrestrial bioluminescent algae, don't necessarily need to be tiny in order to cast a fairly even glow. There's no reason those have to be nonintelligent, friendly, or native to this planet.
Hiding these eldritch swarms from view might benefit everyone's mental health and prevent a panic, and it might also serve to protect the planet's inhabitants in some way. Are these entities also native to mindspace in some way? Can they sense when they're being perceived? Are they attracted to the bright light of intelligence shining out into mindspace?
(Mindspace, in this context, would be the underlying abstract space in which this planet's collective psyche or collective unconscious is built upon - so it's not truly overlapping with the physical world, but each animal or being with a mind connects the physical machinery of the brain with part of mindspace. But, mindspace is an abstract space, so any coordinates in it use a fundamentally different coordinate system, and likely does have any particular relation to our dimensions of space, and possibly even our dimension of time.)
Then again, will these entities remain hidden and harmless forever? ...Did they D E V O U R the Fancy Moon People, who themselves may have dug Too Greedily And Too Deep into mindspace to escape from the entities themselves, and possibly significant parts of the physical moon itself? Are they sniffing around the perimeter of the world, knowing there's something there but being unable to perceive it directly, and therefore unable to navigate through the coordinates of physical space to find their desired prey? (Perhaps due to some effect of the planet's gravity well.) Does breaking through the illusion of the mentally-projected moon remove this protection, and if so... yikes?
Jealous moon deity
You mentioned a moon-based religion in one of your comment responses. An expansionist, interloping lunar deity coming in from another world might want to get rid of the physical moon in order to weaken belief in the planet's native moon deity, while also gathering worshippers among the inhabitants who now see their glorious full moon in the sky. Did they get jealous of their own system's sun shining fully every day, almost always being completely visible except during eclipses? Well, now they fit that bill too, and are even getting worshippers from other solar systems, who can all now gaze at their beauty every single night.
The native moon deity, on the other hand, will presumably remain nearly powerless (being starved of their symbiotic relationship with the planet's inhabitants) until and unless the true moon can be unveiled. (Unless, of course, people can believe in them without having to see them - which could go a C.S. Lewis-style religious fiction route, or it could go another way and frankly discuss whether faith in something for which we cannot directly observe is justified, and what actually qualifies as evidence for that. Could be philosophically interesting either way, especially if the nuance is explored.)
Unhelpful moon entity
The other deities may have hidden the physical moon because of its malign influence. Perhaps the physical moon is a sleeping eldritch abomination, which does not need to wake up or act on the physical plane in order to shed its particular form of uncaring radiance that's notably Bad For Humans. Responsible local creator deities wouldn't want to have that sort of thing messing with their planet and living creations. In the tradition of eldritch horror, such abominations don't have to be evil, intend any harm, recognize living beings at all, or even have intelligence of their own in order to be an apocalyptic threat to living beings.
Cast into the void
A planet that was cast out of its original solar system into the darkness of the interstellar void by a passing massive interstellar interloper (other exoplanets, black holes, destabilization of the solar system's magical weave by The Hubris Of Man, etc.) could have both its moon AND sun provided through mental projection. In this case, in the sun's absence, there would have to be something in place to provide heat, energy for photosynthesis or a suitable equivalent, etc..
However, note that faking a sun would likely be quite a bit more complicated, and require quite a bit more futzing (and magnitudes more powerful magic) to ensure the biosphere didn't utterly collapse. At least in terms of my suspension of disbelief, I could just about believe that one or more incredibly powerful yet still mortal spellcasters could write something into the collective mindscape of a world's inhabitants. However, I'd have a hard time believing that a mortal could use magic to stabilize the heat and energy requirements of a planet's biosphere. That seems much more in the realm of gods, and is a significant anchor of the possible lower limit on their power - it requires not only gods, but POWERFUL gods. And that can be a destabilizing element in itself, in terms of how balanced and believable a system of magic / deities seems.
Weirdly, this also seems plausibly doable by civilizations higher up on the Kardashev Scale, which can use megascale engineering technologies that might be able to effect an entire planet at once. Either way, it seems implausible for mortal entities who only have a single planet to play with to use magic to replace their planet's original sun. Maybe a sufficiently advanced ancient civilization with access to magic and technology could create a self-multiplying and self-perpetuating solution - perhaps the bioluminescent sky entities were created to be a replacement for the sun, and gradually bred to fill up the sky - perhaps they derive sustenance from a source outside of physical space - maybe they get sustenance from mindspace itself, and the ancients accidentally created their horrifying mind-devouring abominations as a last resort to save their planet from an icy death. But, note that breeding that many orbiting, mind-devouring space monsters would take quite a while, so the ancients would have had to have had time to prepare.