Kind of, yes.
That’s its name, basically.
Of course you might say that a lot of planets have moons, and they have distinct names like Ganymede, Europa, and Titan, so why did we never give our own moon a name?
Simply because for most of the time we were aware of it, we didn’t know there were an yother moons in existence, so it sufficed to just call it the moon. The name moon comes from the Old English mona, which could mean moon or month, and can be traced back to the Proto-Indo-European root *me-, meaning to measure, referring to how the phases of the moon were used to measure time.
It wasn’t until 1610 that we found out that there were other moons in existence, specifically the moons of Jupiter, discovered by Galileo. We recognised their similarity to the thing hanging in the night sky, so we called them moons too, but that name had also stuck for our own satellite, so we kept calling it the moon (and it is the only one most of us have any dealings with on a day-to-day basis).
Of course if you want to, you can call the moon Luna, as many do to distinguish it from all those other moons. And that’s fine, though it’s not its sole proper name: rather it’s simply the Latin word for moon. If you wanted to be very fancy, you could call it Selene, which is the Ancient Greek goddess of the moon.
I’ll stick with the moon though. You can’t beat the classics!