This isn't really true. It wouldn't be helpful at all in any emergency situation. This isn't the govt saying they'll hand out money to folks with pets. This is a person being able to deduct a certain amount of money spent on their pets from their overall income at tax time.
SO...in July, your dog is hit by a car and you incur a lot of vet bills, on top of all the regular expenses for your pet. The new tax law won't help a bit at that time. But the following spring, when you do your taxes....IF you're able to take itemized deductions, THEN you can add $3500 (at the most) to those deductions....which, depending on your situation, might lower your owed taxes by $50 or so. Not exactly a bail-out situation for pet owners.