Although we often think of The Taming of the Shrew as being solely interested in the nuances of the 16th-century bourgeois elite, it does much to highlight differences between social classes. In the frame story, Shakespeare goes out of his way to demonstrate the discrepancies between the powerless and lower class Christopher Sly and the noble Lord. Attention to such social disparity carries over into the inset play, where we're asked to recognize the similarities between class hierarchy and the gender hierarchy within the merchant class.
In the play, the Lord's elaborate practical joke emphasizes the social disparity between the nobility and the lower-classes.
The similar circumstances shared by Kate and Christopher Sly suggest that even noblewomen are viewed as second-class citizens.