shrew (n.)

[small insectivorous mammal; malignant woman], Middle English shreue, which is recorded only in the sense of "rascal, evil-doer; scolding woman; undisciplined child;" which is apparently from Old English screawa "shrew-mouse," a word of uncertain origin.

OED calls the word's absence in the "animal" sense from Old English to the 16c. "remarkable." It gives the two words separate entries (2nd ed. print) and speculates that the "malignant person" sense might be original. Perhaps it is from Proto-Germanic *skraw-, from PIE *skreu- "to cut; cutting tool" (see shred (n.)), in reference to the shrew's pointed snout. An alternative Old English word for it was scirfemus, from sceorfan "to gnaw." Middle English Compendium points to Middle High German shröuwel, schrowel, schrewel "devil."

The specific meaning "peevish, malignant, clamorous, spiteful, vexatious, turbulent woman" [Johnson's definition] is c. 1300, from earlier sense of "spiteful person" (male or female), mid-13c., which is traditionally said to derive from some supposed malignant influence of the animal, which was once believed to have a venomous bite and was held in superstitious dread (compare beshrew). Shrews were paired with sheep from 1560s through 17c. as the contrasting types of wives.