Fact: It is possible to see a rainbow at night!
The moon is bright enough to produce a rainbow, and while rare, people do see it.
Since the moon is not near as bright as the sun, the rainbow produced by the moon at night is much weaker than a rainbow produced by the sun during the day.
Our eyes see dim things as black and white, not in colour. So a night "moonbow" will look gray not colourful.