There are a few reasons that this stone got it's name and it would seem its for the same reasons.
When light shines directly into a moonstone, it's unique internal composition creates an effect called adularescence. This is the same illumination effect created by the bright strength of the moon shining through a thin layer of cloud cover.
The stone was called "adularia" in a town of Switzerland that produced some of the first quality mined pieces.
Ancient Hindu legend believed that moonstone was made of solidified moon beams.
There are many legends, myths, and beliefs about moonstone throughout the world. The one thing they all have in common, is that they all associate this stone with the moon- which is almost certainly due to it's adularescent effect.
Some folks prefer their moonshine in a glass, but we prefer it in a stone.