"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"), from PIE *ghans- (source also of Sanskrit hamsah (masc.), hansi (fem.), "goose, swan;" Greek khen; 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). Plural form geese is an example of i-mutation. Meaning "simpleton, silly or foolish person" is from early 15c. To cook (one's) goose is attested by 1845, of unknown origin; 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.
"pursuit of anything in ignorance of the direction it will take," hence "a foolish enterprise," 1592, first attested in "Romeo and Juliet," where it evidently is a figurative use of an earlier (but unrecorded) literal sense in reference to a kind of follow-the-leader steeplechase, perhaps from one of the "crazy, silly" senses in goose (n.). Wild goose (as opposed to a domesticated one) is attested in late Old English (wilde gos).