In the manga, Mu-chan appeared when Yuki used her powers to make Mikuru's "magic lamp" actually magical. He can sometimes be seen as Mikuru's scarf. Whenever he is with Mikuru, she will pet and play with Mu-chan.
Part of the reason he exists is due to Haruhi thinking that, along with rabbits, infinity lions lived on the moon. So, naturally, it came true.