The "issuance decision" and administratrive functions (printing licenses, etc.) are currently handled by different agencies.
In NY State, pistol permits list the individual handguns one may posess. Many counties allow family members to have a handgun on multiple permits - which is the only way you can, for example, have a non-leo family member take your gun if you have a heart attack or car accident and have to go to the hospital.
When you want to buy a gun, you get a reciept, take it to the licensing agency (generally the county clerk upstate), have the gun added to your license and get a purchase coupon you take back to the dealer to complete the transaction. The Rochester, NY gun shows used to, and may still, have a gun licensing booth run by the county clear to do this at the show.
NY just adds a nother license page (they're paper) when one fills up with guns, and a license can have as many pages as a passport once you get enough handguns.
Long guns (outside NYC) are not registered and do not require a permit.