Apparently it's what adds the effect to Lunar Weapons, but it's obsolete since you can just disenchant the weapon and make one yourself on any enchanting table and on any weapon.
EDIT: didn't actually answer the question, but I never actually tried to use the forge during the moonlight.

So yeah, TES games have moon phases, watch out that you make it during the actual moonlight, not just that it's a night time.