She gets laughed out of court. Dependents need a SSN/TIN. The reason for the concept in taxes is because the payer is taking care of the dependent rather than them being a burden of the state.

A dog would not become a burden of the state. At least not permanently. So there is no reason for the state to give someone caring for a dog preferential tax treatment. The state doesn't give a fuck about the dog's role in her life.

The only exception is legitimately disabled people with specifically-trained service animals (which are not considered pets) to deduct some expenses. Again, this is not out of the goodness of the state's heart. It is because it's cheaper than the social services would be.