Hello and welcome to Tractor Forum eagle! I moved your post to the Ford tractor section where it will get more visibility.
The Ford dealer is correct, in most nearly all tractors the transmission, final drive, and hydraulic oil is all one common sump. There are some exceptions. Initially, many equipment manufactures were using GL5 lubricants for the transmission but with the advent of newer lubricants and additive packages went to a common sump lubricant.
Pretty much most of the Ford New Holland equipment calls for 134D lubricant now. Some older owner's manuals on older machines may call for GL5 or other lubricants but have been updated to 134D.
The manufacturer branded oils are much more expensive so a lot of folks go with after market brands. I used John Deere Hygard until it went over $64 per 5 gallon pale. Last service I tried Shell Donax which was about half that price and it has worked very well and I have notice no difference.
I am sure other will jump in here with more to add. Everyone LOVES a good oil thread.