Did a little digging and saw this:
Some experts say that the mug enhances the individual flavors of the ginger beer, vodka, and lime. The copper begins to oxidize when the vodka touches the walls of the mug, they say, which slightly boosts the aroma and enhances the taste of the vodka.
Knowing me, I'd never be able to tell the difference. As long as there is a strong kick to my Mule I'm a happy Muleteer.