The meat of wild goose and domestic goose are very different, just like wild and domestic turkey are different. While domestic goose is notorious for its fat, wild goose is very lean and muscular, and is best eaten while it is young. If, you know, at all.
The term "wild goose chase" is only metaphorically connected to geese: In the 1500s, a popular kind of horse race was a cross-country chase (or pursuit) of a lead horse, who could run in any direction, switching directions at the rider's whim, with all the other horses following its lead. That was a wild goose chase.
The first recorded use of the phrase was by Shakespeare in 1592 in Romeo and Juliet, when Mercutio says, "Nay, if thy wits run the wild-goose chase, I have done...." The current meaning of a hopeless pursuit came later.
Now, aren't you glad you got up and turned on the computer this morning?
Marian Allen, Author Lady