When one transfigures an animal into a water goblet and then back again, has one magically created a water goblet with the soul of an animal? Or does one then have the animal with the soul of a water goblet?
It remains intact.
If the question is asking, which I think it is, whether an animal which has been through Transfiguration can be said to be the same animal the answer is yes. A person or animal gets their memories, personality and identity returned to them when they are re-Transfigured back to their original form again.
We see the principles of Transfiguration at work in human Transfiguration. Malfoy is Transfigured into a ferret and back again with no lasting damage (except to his pride).
A moment later, with a loud snapping noise, Draco Malfoy had reappeared, lying in a heap on the floor with his sleek blond hair all over his now brilliantly pink face. He got to his feet, wincing.
(Goblet of Fire, Chapter 13, Mad-Eye Moody).
Victor Krum also successfully Transfigured his head into a shark's head and back again. Both Malfoy and Krum were returned to their former selves; their souls were unaffected.
Whether animals have souls is less certain. However, the same principles apply to animal Transfiguration as to human Transfiguration (the latter is merely more complex). An animal which has been Transfigured into an object and then returned will maintain its core nature. A dog will still be a dog and retain its doggyness. It's possible to see the retained core nature of an animal in Dean Thomas's partially Transfigured hedgehog.
"Miss Granger remains the only person in this class who has managed to turn a hedgehog into a satisfactory pincushion. I might remind you that your pincushion, Thomas, still curls up in fright if anyone approaches it with a pin!”
(Goblet of Fire, Chapter 15, Beauxbatons and Durmstrang).
The pincushion is clearly no longer a hedgehog but it maintains some of the instincts and dexterity of an animal. That's evidence enough that part of the animal's essence lives on in the pincushion. When Transfigured back you would have the same hedgehog with the same 'soul' (or whatever you want to call it).