Just saying "legal monopoly" doesn't mean much without details.
Licensed black cabs are the only vehicles that are allowed to be flagged down for in London. They're governed by Hackney Carriage laws in the UK and also by some London-specific laws. Getting a license is relatively cheap (no medallion system), but does require passing a test that checks that has questions like 'what is the fastest route from A to B, assuming that it is rush hour and road X has road works?' There are 'mystery shoppers' who audit the taxi system: they flag down black cabs and take rides and, if the driver does not take the most direct route then they can lose their license. The mystery shoppers have varying ethnicities and manners of dress, and refusing to carry one will also result in a loss of the license. Black Cab drivers all know that if they break the regulations requiring them to carry anyone or try to scam a customer, then there's a chance that the customer that they're scamming may have the power to take their license on the spot.
The distinction between taxi and hackney carriage is increasingly irrelevant. I can't flag down a car owned by Generic Taxi Company #47 that's waiting near where I'm standing, but I can call the telephone number printed on the side from my mobile and have the dispatcher tell me that the car next to me is now assigned to me, and then get in. Before mobile phones were widespread, it was very different - you could only call that kind of taxi if you were near a landline (or used a public call box, which added a fairly significant amount to the cost for shortish trips). Uber and other taxi apps are the next step in this. It's now more convenient to press a couple of buttons on the phone than to flag down a passing cab, but the taxi that you get is not regulated in the same way. Uber attempts to claim that their reputation system and pricing model means that they don't need government regulation.
 This has caused some confusion in previous discussions: In the UK, legally speaking, a vehicle that can pick up people who flag them down on the street is called a hackney carriage, any vehicle that carries people for money is a taxi. In common usage, taxi is used for all categories.