You don't need any scientific justification for a good story. The Little Prince lived on an asteroid that would either be unable to hold an atmosphere, or would have such intense gravity gradients that people would be mortally wounded just by standing on it. But we suspend disbelief and enjoy it as one of the most beautiful stories ever. By the way, an anime was made of it in the late 70's and it featured birds flying through space, no explanation needed (the book briefly suggests this as a possibility but never goes into detail). This was even how the prince managed to take off of his little planet, as the birds would use ropes to pull him!
But if you do want some quasi-scientific explanation for how birds can survive and fly in space:
Luminiferous aether or ether ("luminiferous", meaning "light-bearing") was the postulated medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empty space (a vacuum), something that waves should not be able to do. The assumption of a spatial plenum (space completely filled with matter) of luminiferous aether, rather than a spatial vacuum, provided the theoretical medium that was required by wave theories of light.
This idea that space is completely filled with a fluid rather than being a hard vacuum began in the 1600's and ended in the late 1800's. In your world, though, it is true. This is how your moon birds can breath (supposing the aether has oxygen in it) and fly/swim, since they have a medium to push against. The aether may be cold and thin but it never reaches absolute zero nor 0 atm, and your birds have adapted to survive in it.
The aether in your case could be a supercritical fluid, which means it has the properties of both liquid and gaseous fluids at the same time.
Absent a medium to provide lift through wings, your birds would need delta-v budget of a Saturn V rocket, the kind used in the Apollo missions. They wouldn't need to be as massive as the rocket since the payload would be a bird, but still each bird would be as massive as a large house - with most of the mass being rocket fuel and oxygen pressurized into a liquid (which then could be used for breathing as well). For comparison the Saturn V weights a hundred and ninety two metric tonnes with an empty tank, and two thousand, eight hundred and twenty-something hundred metric tonnes when fueled up.