Symbolism, Imagery, Allegory
Petruchio's ridiculous wedding digs are just one kind of disguise we see in the play. Petruchio doesn't normally dress this way – it's all part of his strategy to "tame" Kate by playing the role of erratic and excessively domineering husband. (You can read more about the other kinds of disguises in our discussion of the theme "Art and Culture.") When Petruchio arrives at the wedding ceremony late and dressed like Rainbow Brite's twin brother, he demonstrates his ability to embarrass and publicly humiliate his bride (and her father). Petruchio's outrageous getup makes it perfectly clear that Kate has absolutely no control over what Petruchio wears or how he behaves.
Baptista doesn't have any control over Petruchio either. When the old man asks Petruchio to change into something more appropriate, Petruchio replies that Kate is marrying the man, not his clothes. Petruchio is being a jerk, of course, but there's a whole lot of truth in what he says. Outward appearances are not necessarily indicative of a person's true identity, a lesson we learn over and over throughout the play. This point is lost on Baptista when Petruchio follows it up by saying something like: "By the way, dad, I can't wait for your daughter to wear me out in the sack tonight. Wink, wink."
Petruchio also exercises his control over Kate by controlling what she wears later on in the play. Before the pair set off for Bianca's wedding in Act 4, Petruchio causes a big ruckus over the clothes and hat that have been custom made for Katherine. Petruchio claims they're not up to snuff and insists that he and Kate wear old rags to Bianca's wedding. Kate is livid – like a toddler, she's not even allowed to pick out her own clothes. She's also not allowed to complain about this because Petruchio tells her to zip it, or else they're not going to Padua.