In Act 1, Scene 2, Gremio and Petruchio discuss the latter's plan to woo Kate, comparing her to a lion through the use of metaphor. While Kate is compared to a wide range of animals throughout the play, this particular instance stands out:
GREMIO: But will you woo this wildcat? [...]
PETRUCHIO: Why came I hither but to that intent?
Think you a little din can daunt mine ears?
Have I not in my time heard lion's roar?
Kate is compared to both a wildcat and a lion. Petruchio extends the metaphorical description originally used by Gremio, claiming that a lion's roar is nothing to him and that he is more than up to the challenge of "taming" Kate. Although Petruchio's comment does contribute to the motif of depicting Kate as bestial or animalistic, this particular comparison is more complex. Lions are the apex predators in their environment, capable of pursuing and capturing any prey that happens to cross their path. Gremio simply states that Kate is a "wildcat"; Petruchio is the one who makes the jump from "wildcat" to "lion." It is Petruchio who perceives Kate as the apex predator in her environment: someone with a great deal of promise, capability, and cunning.
In Act 1, Scene 2, Hortensio converses with Petruchio about Bianca. In the course of their conversation, Hortensio reveals his image of Bianca to Petruchio, characterizing his relationship to her through metaphor:
HORTENSIO: Tarry, Petruchio, I must go with thee,
For in Baptista’s keep my treasure is.
He hath the jewel of my life in hold,
His youngest daughter, beautiful Bianca.
The dehumanization of women through object or animal comparison is common throughout The Taming of the Shrew. Hortensio's use of metaphor to compare Bianca to a precious jewel is, therefore, nothing spectacular in the grand scheme of the play. All of Bianca's suitors view her as a prize to be won, as something to possess, rather than a complex human being—again, nothing new. This particular passage, however, reveals more to the audience than just Hortensio's relationship to Bianca. In comparing her to a jewel and stating that her father "hath the jewel of my life in hold," Hortensio acknowledges the connection between the paternal and matrimonial ownership of women. Not only is Bianca objectified by her suitors: she has never not been an object, sitting her entire life like a jewel in a vault, waiting for her father to sell her to the highest bidder.
In Act 2, Scene 1, Petruchio pleads his case to Katherine's father, explaining in great metaphorical detail his plan to woo a woman so resistant to seduction:
PETRUCHIO: Why, that is nothing. For I tell you, father,
I am as peremptory as she proud-minded;
And where two raging fires meet together,
They do consume the thing that feeds their fury.
Though little fire grows great with little wind,
Yet extreme gusts will blow out fire and all.
So I to her and so she yields to me,
For I am rough and woo not like a babe.
Petruchio uses a metaphor to compare both himself and Katherine to "raging fires"—their equivalent passion and twin explosive natures, he believes, will cancel one another out. Petruchio considers himself the "extreme gust" that is capable of extinguishing Katherine's fire. It does not seem, however, that he wishes to crush her spirit in the process; rather, he sees himself as being able to provide her a service, tempering her personality and mellowing her out.
Whether or not this is read as a condescending passage hinges on the performance. One might interpret many of Petruchio's words and actions as calculated performance: after all, he does consciously adopt an exaggerated, shrew-like persona as a scheme to win Kate over. It is equally possible that Petruchio performs masculinity in this passage as a strategy to win over Kate's father.
In the following passage from Act 3, Scene 2, Petruchio uses metaphor to compare Kate to several physical objects, all to describe her as his property:
PETRUCHIO: I will be master of what is mine own.
She is my goods, my chattels; she is my house,
My household stuff, my field, my barn,
My horse, my ox, my ass, my anything.
And here she stands, touch her whoever dare.
I’ll bring mine action on the proudest he
That stops my way in Padua.
Petruchio compares Kate to as many examples of physical property as he can think of: his field, his house, any number of his material goods. On a surface level, this passage reads as incredibly misogynistic. Petruchio appears to buy into the age-old tradition of regarding his wife as his property and treating her accordingly.
Alternatively, one could view this passage as tonally ironic or satirical. Petruchio's overblown and unnecessary declaration almost draws attention to Kate's plight as a married woman—that is, being treated as no more than an object to be owned and passed from father to husband. In line with a central theme in The Taming of the Shrew, Petruchio's speech appears more performative than sincere, comically drawing the audience's attention to the constructed (rather than intrinsic) nature of gender roles.
In Act 3, Scene 2, Gremio—another one of Bianca's suitors—reports gleefully to Tranio that Petruchio is much more of a shrew than his soon-to-be wife. Gremio uses metaphor to compare Katherine to her new husband, arguing that, juxtaposed with his bad behavior, her actions appear tame or even harmless:
GREMIO: Tut, she’s a lamb, a dove, a fool to him!
I’ll tell you, Sir Lucentio: when the priest
Should ask if Katherine should be his wife,
“Ay, by gogs wouns!” quoth he, and swore so loud
That, all amazed, the priest let fall the book.
Shakespeare uses metaphor here, presenting Kate as a "lamb" or "dove" when juxtaposed with her husband, whose behavior at the wedding is shocking to everyone. This comparison highlights just how ridiculous and socially unconventional Petruchio's behavior is. Of course, this is all part of his strategy to win Kate over. Petruchio is an exceptional performer, countering Kate's behavior with his own shrew-like persona. Over the course of the play, Petruchio doesn't just give Kate a taste of her own medicine—he also answers her performance of a gendered literary trope with his own, demonstrating to her the unpleasantness of playing out such harmful roles to their natural conclusion.
In the following passage from the end of Act 5, Scene 2, Kate provides scathing criticism of disobedient women (upon Petruchio's request). She utilizes metaphor in her argument, analyzing what she believes to be the perceived—but in actuality nonexistent—strength of women:
KATE: But now I see our lances are but straws,
Our strength as weak, our weakness past compare,
That seeming to be most which we indeed least are.
Then vail your stomachs, for it is no boot,
And place your hands below your husband’s foot:
In token of which duty, if he please,
My hand is ready, may it do him ease.
Kate acknowledges how ineffective her words and various devices of resistance have been thus far. Her "lances," or methods of attack, are but ineffectual "straws" when placed under patriarchal scrutiny. Where she once perceived disobedience as strength, Kate now considers rebellion against one's husband a form of weakness in women. Kate even commands women who receive her message to "place [their] hands below [their] husband's foot / In token of duty." Troublingly, Kate advocates for full obedience, even in instances where bodily harm is possible. This position marks a drastic change from Kate's outlook at the beginning of the play.