Typically, car insurance is purchased by the same person or corporation that owns the car. The reasoning for this is pretty simple, and mostly has to do with the person who owns the car being the one who has an insurable interest. There are rare occasions when the registered owner and the owner of the insurance policy are not the same, but in most cases the name on your insurance card is going to match the name on your car's registration.
Even if your name is not listed on the insurance card, you still have to be listed on the car's New Jersey insurance policy. The name on the policy is the person who owns the policy and is responsible for making insurance payments or filing claims. In addition to that person, the insurance company will list any other licensed drivers in your household who could drive the vehicle. Your name does not have to be on the insurance card to be covered, but you are not covered unless you are listed on the policy.
The insurance policy is purchased by the person who is financially responsible for the vehicle. Most of the time that person will be the same as the owner of the vehicle, but some exceptions, such as the spouse of the car owner being the primary insured, where it makes sound financial sense for someone other than the car owner to be the one who owns the insurance policy. But the final decision will come down to who is financially responsible for the vehicle, not who owns it, and those can be completely different entities.