The short answer to your question is Yes, its possible to rent cars on either island. Ships brohures always push their tours since they are a huge profit center for the cruise lines. The onboard port lecturers often work for a private company that has a franchise with the cruise line, and they make their money through tour companies and the stores that are recommened on the ships (sorry to be such a cynic, but this is all true). We have been to these islands many times and sometimes do enjoy renting a car in Santorini (check on line for local rental agencies in Fira (or Thira). This being said, many limit their first visit to Santorini to the towns of Fira and Oia..and for this you do not need a rental car (the bus between these cities only cost a couple of dollars). On Mykonos we never rent a car because we think its a waste of money. Mykonos town is less than a mile from the ship and its a place to walk...not drive. We think of the rest of the island as a beach resort and will usually just take a local bus or taxi to one of the beaches. We have never found a reason to drive around the island other than to say we drove around the island.