A good and fun question.
There are several purposes to these puns /word plays. First and most simply, it is fun to watch. It is entertaining to see and hear such verbal facility.
Second, yes, you are right. One purpose is Petruchio taming Kate. Misunderstanding her (for comic effect) opens the door to many of his small victories.
Third, it is a way to establish new and in some ways private communication. They flirt this way—or he pretends they do—and he takes her comments as sexual. Think of that wonderful line, " What, with my tongue in your tail?" He turns her innocent comment into a pretty raw sexual one, letting her know the nature of their relationship (and giving the audience a laugh).
Fourth, and related to all of these, it shows they are in synch. They are meant to be together; this is marked by how they communicate.