I lived in NY for the first 34 Years of my life.. I'm 46 now...was born in 1970.. and moved from Long Island, NY to Melbourne,Florida in 2004.
The way NY used to do its License plates in the 1960's and 1970's (and still does)..... is that they issue new plates and plate numbers when you first register the car just like everywhere else in the U.S. . You can keep those Original plate numbers when they change over to different newer license plate designs, but you have to pay a fee..Usually like $15 or more if you want to keep your old original plate numbers, If you don't pay the fee, The state will just send you New Plates with brand new plate numbers.
Before I moved out of NY, My parents had their same License Plate Numbers on their cars from 1968 to 2001. Even when they bought newer cars, They just transferred the plates to the newer cars and kept the old plate numbers. They paid the extra fee to keep their plate numbers every time. When I moved down here to Florida, I took all of their old 1968 to 2001 license plates. They were all different plate designs spanning over 4 decades, but all the plates had the same two sets of plate numbers for 33 years.
About 4 years ago, I ran into a similar problem that you are having. Even though I knew the license plate number, I DID NOT know the VIN number of my Father's old 1966 Mustang Convertible that he owned from 1968 to 1975. I WANTED TO KNOW THAT VIN NUMBER....BADLY. I wrote letters to New York State, I wrote letters to the GEICO Insurance company, and I got letters back from both of them stating the same exact thing...that Sometime starting in the early 1990's, New York State and also the Geico Insurance Company, only keeps records going back 10 years, and then purges them. I was out of luck.. I'm sorry to say my friend, but you are at a dead end. I'll save you the trouble and just tell you that you'd be better off just to forget it. I know that's not what you wanted to hear, but that's just reality. I fought it, but over time I just learned to accept it. There is no way you will ever get that information unless maybe you are an FBI agent, and even then its an impossible long shot at best.
Melbourne, FL (Formerly from Long Island, NY!)
1965 Mustang Convertible 200cid I6, 3spd Manual
1966 Mustang Sprint 200 Registry Owner/Moderator
1966 Mustang Sprint 200 Convertible 200cid I6, C4 Auto
2007 Mustang V6 Convertible w/ Pony Package