Animal liability insurance covers damage caused by your companion animal to either another person or their property. It does not cover damage to your property i.e. the dog or cat owner’s property, which may be covered by household insurance.
Animal liability insurance may be available under your homeowners insurance policy if it is not excluded. You should check carefully or ask questions for added certainty. Some policies can be hard to understand. You can probably endorse your existing home insurance policy by buying animal insurance liability on top of existing policy. Or you can buy a separate policy protecting you against liabilities caused by your cat or dog.
In the UK I believe that cats cannot expose their owners to a liability to pay compensation for their actions because they are considered free spirits under the law and therefore their owners are not vicariously liable. I am flagging a possibility. On this issue the law will vary from country to country and perhaps state to state in the US.
In the US, as I understand it, animal liability insurance does not cover impounding costs if law enforcement impounds your dog or cat and neither does it cover vaccination costs during the impoundment.
There will be higher premiums probably for certain dog breeds known to have a propensity towards aggression. If you rent a house or apartment you may be able to get renters insurance which covers liability caused by your cat or dog harming an occupier in your flat or their personal property (possessions).
The cost of animal liability insurance will depend in part upon where you live. I’m told that in America owners can be liable for dog bites in some states but not in others. I suspect, too, that certain areas of the country where you live or the city where you live may have a higher incidence of dog bites than in other areas. This increases the risk of a claim being made. The insurance companies will be cognisant of this and increase the premiums accordingly.
Apparently the average dog bite claim in America is valued at $39,017 as at 2018. This seems a very high number but it may be that only claims are made when the damages are serious enough to bother to make the claim.
With respect to incidents where your dog causes injury to a passenger in your vehicle, your car insurance may cover this. As is usual with insurance policies it pays dividends to read the policy from start to finish. Insurance companies have a habit of sliding in exclusions and restrictions which may seem counter to common sense but which are there nonetheless. You should be aware of them.
The bottom line with this sort of insurance is that it’s about peace of mind. If you are the sort of person who likes peace of mind and has a reasonable disposable income and in addition are perhaps a little more risk averse than others, then I would go for it. It’s about personal choice as usual.
Postscript: thinking about it, I would have thought that if you are a person with two pit bull terriers or pit bull mixes living in a smallish house with regular visitors and you are not completely sure about your dogs’ behaviour around strangers then you would be an ideal candidate to take out animal liability insurance!