If you need to obtain your insurance information from the DMV this is usually in response to a lapse of liability order. The DMV issues these as a warning when a car goes uninsured. It is mandated that every vehicle operating on the road have at least liability insurance to legally be used. Checking for insurance information requires this document as well as the 10-digit pin that comes with it has to be entered. You do not even have to be in the building as it can be done online.
Other than that the only documentation needed is the car's VIN number. In most states, including New York, no other documents or forms of identification are needed to check insurance. The VIN number is found at the front of the car in the driver's side portion of the windshield. The 10-digit pin is located at the top of the warning. Accessing information can be done online as both numbers or entered and accepted.
On the other hand, if you want to obtain the insurance of another driver it can be achieved. All you need is their license plate number. The DMV will not provide these details unless you have a valid reason but if one is provided they will allow access to another driver's insurance information. This is usually done in the case of a hit and run, but it can also be accomplished for other purposes as well.