mule (n.1) Look up mule at
"offspring of donkey and horse," from Old English mul, Old French mul "mule, hinny" (12c., fem. mule), both from Latin mulus (fem. mula) "a mule," probably from a pre-Latin Mediterranean language.
The mule combines the strength of the horse with the endurance and surefootedness of the ass, and is extensively bred for certain employments for which it is more suited than either; it is ordinarily incapable of procreation. With no good grounds, the mule is a proverbial type of obstinacy. [OED]
Properly, the offspring of a he-ass and a mare; that of a she-ass and a stallion is technically a hinny. Used allusively of hybrids and things of mixed nature. As a type of spinning machine, attested from 1797 (so called because a hybrid of distinct warp and woof machines). Meaning "obstinate, stupid, or stubborn person" is from 1470s; that of "narcotics smuggler or courier" first attested 1935.
mule (n.2) Look up mule at
"loose slipper," 1560s, from Middle French mule, from Latin mulleus calceus "red high-soled shoe," worn by Roman patricians, from mullus "red" (see mullet (n.1)). Related: Mules.