Sorry to provide a boring answer but the fact is that it is actually the other way around. We have called our moon the Moon for millennia. It was always special, the biggest and brightest celestial object after the Sun. It even occupied its own celestial sphere. No wonder we gave it a name.

Then came the 17th century and we found out that our Moon is just a natural satellite and that other planets have their moons too. It was too late to rename ours so we had to come up with names for theirs.

Bot our Moon is special after all. There is no other planet we know of that has a moon so large in relation to its host planet's size. Pluto's moon is larger, to the point that Pluto and its moon orbit a common centre of gravity that is outside the Pluto's interior. But sadly, Pluto is no longer a planet so our Earth-Moon system is unique again.