What kind of license / certification will be required to own / fly one of these? Presumably something less than a PPL, otherwise the barrier to entry will be too high.
Short answer: You will almost certainly need a PPL at least.
On the subject of licensing for flying cars, nobody really knows, because no company has even come close to bringing one to market - and the term "bringing to market" for a flying car involves all the legalities of certification and licensing.
The U.S. based Chinese-owned firm Terrafugia and AeroMobil are also attempting to develop a two-seat flying car called the Transition, however, safety and development of the law for flying cars are just two of the challenges that must be met before they become a reality and not just a novelty. The Transition is designed to operate both on land and in the skies.
In terms of aviation certification the Aeromobil specifications state
Aerospace: CS 23 Rev 5/ELA 1 (EU) & 14 CFR 23 Rev 5 (US)
That means for the US it is certified under Part 23 which will almost certainly need a minimum of a private pilots license. In the EU it is pretty much the same deal as many of the aviation standards/certifications around the world are based on ICAO norms.
Not that you asked for it, but in my opinion the only likely route to market for personal flying vehicles is ones where you're not doing the flying. One day I'm sure we'll have something like an automated Uber which ferries you A to B short distances. But something where you drive and fly is hugely unlikely.
In the eyes of the FAA, it’s considered an Airplane, Single Engine Land category and class of aircraft. There is no license required to own the vehicle in the United States, save from registering it in your state of residence.
If you intend to operate it an an aircraft and serve as pilot in command (PIC), you’ll need to hold, at a minimum, a Recreational Pilot Certificate with Airplane Single Engine Land ratings. This would allow you to at least fly the thing but imposes severe limitations on where and how you can travel in it. For any practical use of the Aeromobil as an aircraft, it is recommended that you hold at least a Private Pilot Certificate with Airplane Single Engine Land and Instrument Airplane ratings. This would allow for unrestricted operation of the aircraft as well an conducting Part 91 flight operation under IFR. You will also need to hold at least a Third Class Medical Certificate, or have held a valid FAA medical anytime after July 15, 2006 and conform to the guidelines of Part 68 for BasicMed.