The most we know about Mugetsuís relationships are, that before it was in Ame-Warashiís hands, and then it was handed as payment to Yuuko. While it has a normal relationship with the witch, there is not much to say about it, Mugetsu does like Yuuko, but the one it loves, is and always will be Watanuki.
The relationship with the boy is somehow interesting and funny on some occasions, especially at the beginning, when we see how the small fox gets attached to him. Some say itís because he attracks evil spirits, so the fox follows him, just because. I on the other hand think, that Mugetsu genuinely loves him. Like love at first sight. It loves to be around him, and caress him and show him itís love. Itís protective when Watanuki is in danger. And even if the boy has his classical anger issues at the beginning around Mugetsu, because, sneaking in his clothes to go to school with him, itís a little exaggerating. I think there is mutual love. Itís Mugetsu who goes with Watanuki on the rescue mission to save Zashiki-Watashi from the nets of Jorougumo (the spider-woman-deamon). And later, itís also shown how much the fox loves the black haired boy, when it begin sulking, wanting to be named, and Yuuko comments on the fact, that is normal that we want to be named by the person one loves the most.