Can you copyright an animal?

Not as such.

If you draw, paint, sculpt, or otherwise create the likeness of an animal, your work is automatically protected by copyright.

If you managed to create a new kind of animal, it may be possible in some jurisdictions for you to patent the animal.

If you owned, say, a dog with a very particular, unique, and/or recognizable image, it could be possible for you to trademark the image of that animal, particularly in a specific pose or setting, such as the fox terrier 'Nipper', who appears in the famous RCA-Victor trademark of a dog listening to a Gramophone, with the accompanying trademark phrase "His Master's Voice".

But, you can't 'copyright', for instance, your dog Pepper for normal purposes... Which probably won't be that great an inconvenience, as it would be beyond the means of current technology for anyone to make and distribute unauthorized copies of Pepper without first going to a great deal of effort and expense, and probably requiring your (and Pepper's) cooperation.