I mean, does it matter if it's full coverage, no medical coverage, liability, or what not? Isn't an auto insurance suppose to cover any bodily injury no matter what party?

Insurance is something you want to be careful about when buying. All states have different requirements for what must be covered. With that being sad, not all policies have to have bodily injury for everyone involved. Also, state minimums are very low. MA's requirement is 5k per incident. Put 4 people in a serious car crash and 5k doesn't cover much.