Is pet insurance for if your poodle tears up your curtains or your show champion poodle gets hit by a car?
Insurance gets complicated, and there's always some insurance company that does things differently than the others.
For what I've read about, this is what usually happens:
If your dog tears up your curtains, you buy new curtains with your own money. No insurance involved.
If your dog gets hit by a car, pet insurance might pay the vet bills for the dog (read the terms, because you probably have to pay part of it too).
But the car insurance of the driver or owner of the car could be involved, too. Car insurance usually pays something if there is "property damage" caused by the car, and a dog is "property" just like a fence or mailbox that a car might run into.
If your dog bites someone in your own family and they need medical care, the family medical insurance policy covers it.
If your dog bites someone else, their medical insurance probably pays the doctor bills, but then their insurance company sues your homeowners insurance company. Some companies will not sell you homeowners insurance if you have a dog of certain breeds, because they are afraid of exactly that.