I do not see anything indicating that it is limited slip, so a std gear oil should be fine. No special additives required if not limited slip. A synthetic would be good, like 75W-140 or similar, because you will generate some heat with the load.
If you have an overall picture of the housing, especially the cover that can help tell what model it is. May also have a stamp on the axle tube.
2005 Kenworth Showhauler truck conversion.
. 44 ft, Cummins ISX15 450 hp, 10 speed, twin screw
I used to have a handle on life, but it broke