There are a few reasons that this stone got its name, and it would seem that it is the same reasoning.
When light shines directly into a moonstone, its 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 all of them associate this stone with the moon- which almost certainly is due to its adularescence effect.
Some folks prefer their moonshine in a glass, but we prefer it in a stone.