Trillman is a Golden Retriever. Like many familiars, he was once a warlock himself. Although he is strict with DWW about following the rules, he once broke one of the Other Realm's biggest rules, which is why he is now spending time as a talking dog.
Unlike most Golden Retriever's, Trillman is not the most patient dog. As he struggles to teach DWW magic he begins to realize that DWW is lacking in responsibility. He only knows how to teach magic, raising kids isn't something he was ever taught to do. Not knowing how to teach a challenging young witch like DWW, Trillman quickly begins to lose patience with her.
Ironically, Trillman had wanted to become a teacher, but failed at this. Ultimately, it's not because of DWW that he quits, but because of his own past failures. Frustration with himself led to much of his impatience with the young witch.
DWW and Trillman belong together, but it's not until they spend much time apart will they come to realize this.