From shrew +‎ -ish. Compare Middle English schrewis, shrewessh, shrewyssh (“wicked, malignant”). 

For Shakespeare's early modern audience, a shrew, or a shrewish person, was someone who lacked self-control, or who posed a threat to others. Simply put, someone or something in need of taming. 

The shrew – an unpleasant, ill-tempered woman characterised by scolding, nagging, and aggression – is a comedic, stock character in literature and folklore, both Western and Eastern. 

The earliest known use of the word "shrewish" was in the Middle English period (1150—1500). The Oxford English Dictionary (OED) has evidence of the word from around 1480 in St. James the Great.