The person is covered not the vehicle, that's why it doesn't matter what car you are in. If someone borrows your car they are not covered unless you go to the car so the member is there with ID.
AAA drivers are never exclusively AAA but are the exclusive company for a given area by contract. If the contracted company can not get to the tow they can kick it back and AAA will dispatch the next closest AAA company or call around to find one to do the job if no other is available. The contracted company has exclusive right to any service in their contracted area though. AAA does not have any tow trucks of their own they simply contract existing companies for a given area. They have specific boundries to the area they cover. AAA does not pay enough for a company to be exclusive to AAA. I used to be a driver for an AAA company and had to know all the ins and out of it. There is even more to it if you want to get into abuse of the system, non-payment of overages, etc... Speaking of payment, most AAA companies will take a check from an AAA member because AAA will cover the check up to a certain amount. (don't remember the amount anymore)