Muta is a supporting character in The Cat Returns.
Brash and grumpy, Muta is known for his unchecked sense of sarcasm and condescension towards people and his other companions, and for his huge appetite. This accounted for his chagrined relationship with Toto, whom he has a mutual dislike with, though they wouldn't hesitate to team up and work together when the need arises. In spite of his seeming indifference, he did eventually warm up to Haru, whom he praised for standing up for herself and later aided in her escape from the Cat Realm.
Whisper of the Heart
Although the character design is very similar, in Whisper of the Heart Muta is known mostly as "Moon" by Shizuku and Seiji, and is only called "Muta" once at the end of the film by an unnamed little boy speculated maybe to be one of Muta's owners. Much like in The Cat Returns, Muta's purpose in this film is to lead the main character, Shizuku, to the characters who will help her overcome her challenges and grow as a person (these characters being Seiji and his grandfather, Mr. Shiro Nishi).
Although The Baron is introduced in this movie and plays a pivotal role in both the lives of Mr. Nishi and Shizuku, his role is a symbolic one, as he remains an inanimate object throughout the film only coming to life in Shizuku's book. Muta, as he is depicted as a more average cat, doesn't appear have any connection to the Baron in this film/doesn't acknowledge him.
The Cat Returns
Muta was given a supporting role in The Cat Returns, where he is depicted as The Baron's portly companion at the Cat Bureau. A mysterious voice instructed Haru to look for Muta at the Crossroad where he frequents, who leads her to the Bureau to meet the Baron. Soon after meeting them, Haru and Muta are forcefully taken to the Cat Kingdom, leading Toto and the Baron to go after them.
Inside the castle, Haru and Muta are treated to a royal banquet, but Muta is suddenly trapped in a pitcher of catnip jelly; he eventually escaped in the ensuing fight when the mysterious visitor who danced with Haru is revealed to be the Baron himself. He, Haru and the Baron were then aided by Yuki, a servant cat who initially warned Haru to leave the Cat Kingdom.
As they were confronted by the King, Muta reveals that he is actually Renaldo Moon, a notorious criminal who gained a reputation for devouring a whole lake of fish at one time. Muta praises Haru for standing up for herself after rejecting the King's proposal to marry him, and helps the Baron and Haru escape from the cat realm.