As a "private organization", Uber of course doesn't have to do anything.
But if it wants to operate a business on NYC's streets on a (massive) scale -- well, that's a privilege, not a right. And if they want access to that privilege, they're gonna have to play by the rules as set forth by the city's voters and their elected representatives. Who have a vested in interest having reliable access to that data for well, a whole bunch of pretty obvious reasons.
It's called "rule of law", a concept which Uber has demonstrated considerable difficult in understanding thus far. Once they do, we can perhaps have a conversation about whether certain regulations are really useful or necessary or not. But it needs to be based on the pragmatic merits (or lack thereof) of those regulations. Arguments on the basis of "they're a private organization; you have not right to tell them what to do" just don't hold a lot of water in these contexts.
This is a slippery slope.
No, it's just life in the big city. And it's about time Uber got used to it.