Uber/Lyft competes less with Manhattan taxicabs than they do with other legacy FHV sectors like black car, limo, livery, and commuter vans.
Each for-hire vehicle sector has its own host of regulations, and the Uber/Lyft business model to undercut everyone (in addition to relying on part time labor) was to claim they were intrinsically different and none of the existing reg schemes applied.
The New York City Council created a new FHV license class I believe in 2018 for "high-volume" services which allows the City to impose regs solely on Uber/Lyft. They also enacted a license moratorium at the same time because Uber/Lyft had grown out of control and congestion was at its peak, but the moratorium actually benefitted the high-volume companies at the expense of the other FHV sectors because they could just poach drivers/licenses and the other sectors couldn't afford to do anything about it.