Goose is native and ancient, from Old English gos, with cousins across Indo-European: Sanskrit hamsah, Greek khēn, Latin anser, Lithuanian žąsis. The word is probably imitative of the bird's honking, which explains why so many unrelated branches of the family have something so similar.
Two sound changes are visible in the modern form. The n that other languages kept — as in German Gans — dropped out before the s, which is regular in English and also explains tooth beside German Zahn. And the plural geese is an example of the old Germanic vowel change called i-mutation, the same process behind mouse and mice, foot and feet, man and men. Those plurals are not irregular so much as a fossil of an older system, older than writing in English. Johnson, defining the bird, could not resist noting it was "proverbially noted, I know not why, for foolishness."
goose (n.) "a large waterfowl proverbially noted, I know not why, for foolishness" [Johnson], Old English gos "a goose," from Proto-Germanic *gans- "goose" (source also of Old Frisian gos , Old Norse gas , Old High German gans , German Gans "goose"). This is reconstructed to be from PIE *ghans- (source also of Sanskrit hamsah (masc.), hansi (fem.), "goose, swan;" Greek khēn ; Latin anser ; Polish gęś "goose;" Lithuanian žąsis "goose;" Old Irish geiss "swan"), probably imitative of its honking. Geese are technically distinguished from swans and from ducks by the combination of feathered lores, reticulate tarsi, stout bill high at the base, and simple hind toe. [Century Dictionary] Spanish ganso "goose" is from a Germanic source. Loss of "n" sound before "s" is normal in English (compare tooth ). The plural geese is an example of i-mutation . The meaning "simpleton, silly or foolish person" is from early 15c.; proverbial expressions indicating "know no more than a goose" are by late 14c. The bird's reputation for stupidity is not classical. Roman geese were sacred to Juno and held in highest honor. In Athens (as also in Rome) they were noted for watchfulness, and as an erotic bird. Tales told of various geese enamored of a boy, a physician, a philosopher (see Thompson, "Glossary of Greek Birds"). It was a term of endearment in later Greek (compare duck (n.1) as an Elizabethan term of endearment). To cook (one's) goose is attested by 1845, of unknown signification; attempts to connect it to Swedish history and Greek fables are unconvincing. Goose-egg "zero" is attested by 1866 in baseball slang, from being large and round. The goose that lays golden eggs (15c.) is from Aesop. In Homer as in Middle English typically with the epithet "gray" ( argos ). Thompson notes that "The Geese in the Odyssey are tame birds, ... in the Iliad always wild." goose (v.) "jab in the rear," c. 1880, from goose (n.), possibly from resemblance of the upturned thumb to a goose's beak, or from the notion of creating nervous excitement. Related: Goosed ; goosing . In 19c. theatrical slang, to be goosed meant "to be hissed" (by 1818), hence, probably, bird (n.3), the rude gesture. However the goose has had an erotic reputation since ancient Greece (see the noun). A broad range of English sexual slang senses historically cluster around goose and gooseberry ; goose and duck was rhyming slang for "fuck;" Farmer identifies Winchester goose as "a woman; whence, by implication, the sexual favor," and goose as a verb "to go wenching, to womanize, also to possess a woman." He also has goose-grease for a woman's sexual juices, while gooser and goose's neck meant "the penis." Gooseberries (they are hairy) was "testicles," and gooseberry pudding "a woman." also from c. 1880