E-bikes are motor vehicles and can't be used on bike paths in NY state though they are street legal if registered. NYC park police actively ticket E-bikes on the west side.

In California, I'm fairly certain that if they're restricted to below 20mph they're just fine in bike lanes/paths.

Correct. https://espinbikes.com/wp-content/uploads/2017/07/CA-E-Bike-...

Do you know if the restrictions apply to the electronically assisted "e-bikes". They pretty much operate like a normal bike, but require minimal effort to go up hills.

Example: https://www.limebike.com/electric-assist-bike

ebikes are pedal assist bikes. If there is no pedal assist, and the motor can be activated with a throttle, then it's an e-motorcycle.

The presence of a motor makes it a motor vehicle in NY. Electric assist bikes are no different functionally or legally from gas powered mopeds.

How would one ticket an ebike since they have no registration or license numbers?

You ticket the driver.

I do know about this restriction. You can get your physician to write you a note stating that you need an assisting device for locomotion purpose and show this to the officer trying to site you.