The privacy of the driver is outweighed by the need for the public to feel they will safely be taken to their destination in an enclosed vehicle totally controlled by another party.
https://data.cityofnewyork.us/Transportation/Medallion-Drive...
In any case, rules are different for different jurisdictions and entities. If NYC TLC did have that requirement, then it's possible that real name display is mandatory for Lyft drivers operating in NYC, given that TLC has some oversight over services like Lyft. However, TLC also has some say over livery cabs. It's been awhile since I've been in a livery cab, but I think they do operate under different requirements in terms of what's required for display.
You can definitely argue that they should show their real name, but it is not a good argument that because taxi requires it, lyft should as well.