Both plays deal with the difficulty and stuffiness of the social structure. In "Much Ado About Nothing," the characters often have to give way to people in higher authority roles. Benedick notes frequently that others in the household speak in lofty words to seem of a higher position than they are.
In "Taming of the Shrew," Bianca is not allowed to marry until her older sister does, and Katherine does not wish to marry. They are forced to accept their roles as upper middle-class society ladies, however.
In both plays, the motif of women as animals that must be tamed runs throughout. Beatrice says she will tame her "wild heart" to Benedick's loving hand, which refers to the old practice of taming falcons. And of course, Katherine is the titular shrew of "The Taming of the Shrew." Petruchio says he will tame her like a falcon as well.
Both works feature acts of public humiliation. In "Much Ado About Nothing," Hero is denigrated at her wedding ceremony, made to seem like an adulterer in front of everyone. In "Taming of the Shrew," Petruchio often criticizes Katherine in front of people and even shows up to their wedding dressed like a clown in order to humiliate her.