Why do all the other planet's moons get to have names, but Earth's moon does not?
Tradition, going back to the time (before Galileo's discoveries) when our moon was assumed to be the only moon. Similarly, the sun has no name either, unlike all the other prominent stars.
For those who seem to need a name for the moon, it's often called Luna, which is "moon" in Latin. When your grandchildren are living on Mars, presumably they won't call Earth's moon "the moon." Until then, there's not much room for confusion.