Unless you've been living under a rock, you'll know that the Stanford Axe is awarded to the winner of the Big Game, or in recent history, to Stanford. It is the Stanford Axe, after all.
The winner of the oldest rivalry game in the West now gets the Axe basically because Cal finally won a baseball game, which in April of 1899, was apparently cause for riot. Per Cal alum Sean Rouse's report for the Axe Committee (which all the facts for this article come from), Stanford was one of the top-ranked baseball teams just before the turn of the 19th century.