Taxi medallions are a license to operate taxis in some of the major cities across US - New York, Chicago, Boston - to name a few. Back in the Depression days (1930s), cities were desperate to raise cash and bring in new business - and thus began offering taxi medallions (i.e. the license to operate cabs) through auctions. The rich men and women of the day bid and won several medallions and began operating cabs in those cities. This brought in some much needed new business at the time (and jobs) while raising cash for the city.
Well, the depression ended, but the system has remained in place. There have been several studies undertaken with some indicating that the system is broken, with others showing that all is well. The one aspect of the system that has definitely made the owners of the medallions quite rich is the limited number of medallions that exist in the system (it takes an act of the government to issue more medallions) and the auction system that consistently bids the prices higher. New York City is looking at medallion prices in the $400,000 range. Obviously, this implies that the actual cab driver has a long way to go before he (female cab drivers are rare) can hope to own a medallion and keep all his earnings to himself.
Thus if you ask "Who owns NYC taxi medallions" - the answer is - private citizens such as you and me, who are willing to put up the money to own the medallion, and are willing to take care of that responsibility.