How is The Taming of the Shrew similar and different to the story "The Sleeper and the Waker" from The Arabian Nights?
The ways these are different are pretty well obvious. One is an English drama written by a poet. The other is an Arabic story that comes to us in English translation as prose with a few quotes of poetry in it. The one tells of Western experience of relationships between women and men, which we call universal, though, in other cultures, the universality may not carry over. The other is about (1) relationships between men and other men and the role of hospitality and (2) relationships between men and their money, a theme that also applies to Western cultures. The language is very different one from the other. In the English one, the language is high diction with hardly a superfluous word (unless, of course, you agree with Dr. Samuel Johnson ...), whereas the Arabic is told in middle diction of conversation.
It blots thy beauty as frosts do bite the meads,
Confounds thy fame as whirlwinds shake fair buds,
And in no sense is meet or amiable.
A woman moved is like a fountain troubled, (Taming of the Shrew)
When the Caliph heard these his verses and saw how apt he was at couplets, he was delighted with exceeding delight, and taking the cup, drank it off, and the twain ceased not to converse and carouse till the wine rose to their heads. ("The Sleeper and the Waker")
One of the similarities is that both the English and the Arabic works are about power and authority and control. In this frame of reference, another similarity is that the weaker characters, Katharina and Abu al-Hasan, struggle to gain control in the face of greater power. Another is that the more powerful ones--Petruchio and the Caliph, Commander of the Faithful--use trickery to gain the advantage (as they each saw it) over the weaker ones. This leads to the most significant way in which the English play and the Arabic story are similar: in the end, the weaker, subservient one in each learns and applies the powerful ones' techniques in order to claim and assert power of their own.
Quoth [Abu al-Hasan], "I wrought this sleight so I might get of thee the hundred dinars and the piece of silk, and all this is an alms from our lord. But now make haste to give me the thousand dinars and acquit thee of thine oath." ("The Sleeper and the Waker")
Forward, I pray, since we have come so far,
And be it moon, or sun, or what you please:
An if you please to call it a rush-candle,
Henceforth I vow it shall be so for me.
Then, God be bless'd, it is the blessed sun:
But sun it is not, when you say it is not;
And the moon changes even as your mind.
What you will have it named, even that it is;
And so it shall be so for Katharina. (Taming of the Shrew)