goatish adj.

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]

adjectival word-forming element, Old English -isc "of the nativity or country of," in later use "of the nature or character of," from Proto-Germanic suffix *-iska- (cognates: Old Saxon -isk, Old Frisian -sk, Old Norse -iskr, Swedish and Danish -sk, Dutch -sch, Old High German -isc, German -isch, Gothic -isks), cognate with Greek diminutive suffix -iskos. In its oldest forms with altered stem vowel (French, Welsh). The Germanic suffix was borrowed into Italian and Spanish (-esco) and French (-esque). Colloquially attached to hours to denote approximation, 1916.

The -ish in verbs (abolish, establish, finish, punish, etc.) is a mere terminal relic from the Old French present participle.