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.
Which makes sense. There's likely a lot of drivers who want to go by "Tom" instead of "Thomas" or whatever their full official name is – which is even more important for drivers with very uncommon names who want to go by something more familiar/pronounceable. There's probably a decent safety argument for not requiring drivers to have their full real names be accessible to the user, especially in cases where a driver has an uncommon (i.e. easy to doxx) name.