From what I was told by the DEC, all I had to do was register my pond, and in return I was given the right to fish any time I wanted, and stock any fish I wanted from a registered fish dealer. My only restriction was to only stock fish from a licensed dealer of fish in NY.
I did not register as a fish farm, only as a private owner. Perhaps that is the difference? Are they assuming you are raising fish for sale at stores and not for personal use? Maybe it is a pond size restriction? I know that if I went over a 1 acre pond, I would have had to jump through hoops, but I thought that was mostly for design concerns on the dam and overflow structures.
NYS may be a pain in the rear end for its nannying, but I don't think it is that bad. I would call the department and get some clarification, as I think you are getting the wrong licenses for your pond.