Married, as in legally binding... we already sort of can, it's just called ownership. My cat is chipped, the chip has my name, and my address, it says my cat is my cat, it's a sub-dermal wedding ring. I made a commitment with the shelter i got him from that i will care for him until death do us apart.
If i want to do a little ceremony, invite some friends and family, make a vow and eat cake, I am also able to do that as I desire.
So what we're left with is the legal implications of what a marriage is. To my understanding, it's mainly about property. Animals do not own property. It's a merging of assets, animals do not have assets. A merging of certain societal obligations, like taxes, benefits, debts, etc. animals have none of those either.
So when discussing the consequences of animal marriage, they are non-existent, to my understanding. The question then becomes not "should it be allowed?" but rather "what would be the point?" No legal interference is needed, except when it involves animal cruelty, which is a punishable offense regardless of your status relative to the animal. So a legally binding marriage between man and animal would be... a waste of time.