In New York, taxis are part of the public transportation network. Taxi drivers go through extensive training to know where pretty much any address in the five boroughs is, and the fastest routes to get there. In order to incentivize taxi drivers to go through this training, and to incentivize owners to buy and maintain vehicles, the city limits the number of vehicles on the road at a given time to incentivize drivers and owners to go into the taxi business. According to the 2014 NYC Taxicab Fact Book, there are 13,437 medallions in circulation -- assuming that 70% of them are on the road at any one time, and taking the figure that 94% of pick-ups occur in Manhattan or at JFK or LaGuardia, that implies that there are over 200 taxis operating per square mile in Manhattan ... that's A LOT !!! It's not uncommon to grant a limited monopoly to a public utility to ensure consistent service.
Why lease medallions? Utilization. It would be really hard (and unsafe, and unpleasant) for an owner-operator to be in the car for more than 80-90 hours per week, whereas a leased car can be in operation for almost the entire 168 hours.
All that said, driver knowledge is much less important than it once was, with GPS (though taking routing advice from a mapping service in Manhattan could be a bad idea). Safety is probably less of a concern now too, since Uber tracks its drivers' every move. And, it's clear that medallion owners are reaping a lot of profit at the expense of drivers willing to work for sub-minimum wages. So, it's probably time to renegotiate the contract.