Is it correct to say "The dog plays with its bone" or "the dog plays with his bone"? Are both correct and why?
Yes they are both correct. See explanation.
If we talk about any dog, we use the preposition it, but if we have some connection whth an animal we are talking about (for example we are talking about the dog we love; not necessary our dog) we could treat it like a person and use the preposition he or she depending on the dog's sex to accent that the dog is important to us in some way.
This manner can be used not only to animals. Seamen usuallly use preposition she when talking about the ship on which they are working.