You can marry an animal, sure. But it won't be an official state-sanctioned or recognised marriage for the sole reason that animals are not natural persons (ie humans) or corporate persons (ie a company). They are natural entities, which have fewer rights and obligations placed on them by society and the legal system because they are unable to perceive higher level functions and concept like "marriage" and "consent", and things like that. Natural entities are not sapient - ie have the ability to reason, be aware of self, and perform logic functions independently, which natural persons can do.
For this reason, natural entities are under the care and supervision of natural or corporate persons (ie the reason why you are liable for neglecting to care for your dog, and not the dog itself).

(On youtube video about homestuck)
Person in video: So Homestuck is probably going to end in the next few months...
(I scroll down to see when the video was made)
2013
(I leave video)