Puppy Chews are a new type of treat introduced in Nintendogs + Cats. They are bone shaped chews that can be unlocked and bought at BARC in the Toy Poodle version by having 2,350 Owner Points or by playing 12 days in a row. You can also get them by walking with the Pedometer. the packet says "Dog Gum", so that is probably what this is.
3DS: "These rich, tasty chews are really satisfying to gnaw on."