Another consequence is that on the Libby app you have better chances to find something available, and each one’s limits are separate!

The tbree systems in NYC also serve the entire state, not just the city. Any NYS resident can get a card at each. I do and use the e book systems for all at the same time even though I rarely go into the city these days.

Seems that even as three separate systems, they'd at least just let a card from any of them work at all three. Or issue a single unified card and let that be the only point of commonality between them, if all other operations being independent is important to them

Kind of like how many regions will have a unified transit payment card, even if it covers a half dozen completely independent adjacent transit systems

You can also get Buffalo/Erie County.

It seems that NY has a law that allows anyone in the state to be a member at any library in the state, but there's an opt-out if that would put an undue burden on the local system.

In practice, that means anyone in NYS can be a member of any of the NYC libraries + Buffalo. The others have either opted out, require you to visit in person, and/or pay a non-resident fee.