Old English gat "she-goat," from Proto-Germanic *gaito (source also of Old Saxon get, Old Norse geit, Danish gjed, Middle Dutch gheet, Dutch geit, Old High German geiz, German Geiß, Gothic gaits "goat"), from PIE *ghaid-o- "young goat," also forming words for "to play" (source also of Latin hædus "kid").
They are sprightly, capricious, and wanton, and their strong odor (technically called hircine) is proverbial. [Century Dictionary]
The word for "male goat" in Old English was bucca or gatbucca (see buck (n.)) until late 1300s shift to he-goat, she-goat (Nanny goat is 18c., billy goat 19c.). Meaning "licentious man" is attested from 1670s (hence goat-milker, name of a bird formerly believed to suck the milk from goats at night, but also old slang for "a prostitute," also "the female pudendum"). To get (someone's) goat is by 1908, American English, the source of many fanciful explanation stories; perhaps from French prendre sa chèvre "take one's source of milk," or more likely it is "to steal a goat mascot" from a racehorse, warship, fire company, military unit, etc.
... to become separated from your goat is a thing no soldierman is willing to contemplate. ["Letitia, Nursery Corps, U.S.A.," in American Magazine, vol. lxiv, June 1907]
-y 2
adjective suffix, "full of or characterized by," from Old English -ig, from Proto-Germanic *-iga- (source also of Dutch, Danish, German -ig, Gothic -egs), from PIE -(i)ko-, adjectival suffix, cognate with elements in Greek -ikos, Latin -icus (see -ic). Originally added to nouns in Old English; used from 13c. with verbs, and by 15c. even with other adjectives (for example crispy). Adjectives such as hugy, vasty are artificial words that exist for the sake of poetical metrics.
goatee n.
"pointed tuft of beard on the chin of a shaven face," 1844 (as goaty; current spelling by 1847), from goaty (adj.). So called from its resemblance to a male goat's chin hairs.