Puppies who gnaw on shoes are probably doing it because they’re teething and the soft rubber or leather feels pleasant. Because puppies find relief, they tend to make a habit of it.
For adult dogs, however, the motivation is more varied. For one thing, shoes are easily accessible. They’re usually found at ground level, making it easy for a dog to case the scene, grab the shoe, and sprint off. From there, shoes become a kind of sensory experience. They have their owner’s scent as well as smells that have been transported from walking in other areas. It’s kind of a dog travelogue, with the pet able to explore where someone has been. The more pungent, the better.
Dogs might fall asleep next to a shoe for the same olfactory reason. Because it smells familiar, it’s comforting.