When you equip a mount, your pet hops into your backpack for the ride. They cannot move as fast as your mount, and therefore don't appear when you have your mount equipped.
Just like a hat, you equip a mount (and remember you have to be in an outdoor location to equip a mount) by either double clicking on it in your backpack or single clicking on the mount in your backpack, and then clicking the equip (hand) icon.
