"a large waterfowl proverbially noted, I know not why, for foolishness" [Johnson], Old English gos, from Proto-Germanic *gans- "goose" (cf. Old Frisian gos, Old Norse gas, Old High German gans, German Gans "goose"), from PIE *ghans- (cf. Sanskrit hamsah (masc.), hansi (fem.), "goose, swan;" Greek khen; Latin anser; Polish gęś "goose;" Lithuanian zasis "goose;" Old Irish geiss "swan"), probably imitative of its honking.
Spanish ganso "goose" is from a Germanic source. Loss of "n" sound is normal before "s." Plural form geese is an example of i-mutation.
Meaning "simpleton" is from 1540s. To cook one's goose first attested 1845, of unknown origin; attempts to connect it to Swedish history and Greek fables have been unconvincing. Goose egg "zero" first attested 1866 in baseball slang. The goose that laid the golden egg is from Aesop.