That is for the "black cab" taxis, the ones you can hail by waving on the street.
For a private hire vehicle (the kind you telephone or use an app to book), the price is under £700 (under £500 if the driver speaks English).
That doesn't seem unreasonable. The GP comment was referring to the American system where the license are traded at very high costs (tens, hundreds of thousands of dollars).
https://tfl.gov.uk/info-for/taxis-and-private-hire/licensing...