I'm going to nerd out for a sec. You've been warned.
It's due to the fact that moons only reflect sunlight; they don't emit light of their own. Since the moon is just a reflecting surface, the color of the surface doesn't really have an impact on the color of the reflected light - light comes in, bounces off the surface, and hits your eyes. So, really, only the source color matters - if Nirn's sun was red, then you'd see red moonlight. Earth gets white moonlight not because our moon is white (that's just a coincidence), but because our sun emits white light (even though its peak emissions are in the green part of the spectrum).
Anyway, to test this for yourself, shine a flashlight on surfaces of various colors, and then look at the light itself, not the surface. You'll see that the light from the flashlight that you see doesn't change colors depending on the color of the surface it's reflected from.
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It's due to the fact that moons only reflect sunlight; they don't emit light of their own. Since the moon is just a reflecting surface, the color of the surface doesn't really have an impact on the color of the reflected light - light comes in, bounces off the surface, and hits your eyes. So, really, only the source color matters - if Nirn's sun was red, then you'd see red moonlight. Earth gets white moonlight not because our moon is white (that's just a coincidence), but because our sun emits white light (even though its peak emissions are in the green part of the spectrum).
Anyway, to test this for yourself, shine a flashlight on surfaces of various colors, and then look at the light itself, not the surface. You'll see that the light from the flashlight that you see doesn't change colors depending on the color of the surface it's reflected from.
Okay, I'm done.