Mobil DTE is just an example, the Harvest King or similar hydraulic oil of the same ISO rating is fine. Typically hydraulic oils have anti-foaming and anti-rust inhibits which you want. You do not want extreme pressure (EP) agents which are more commonly found is differential/hypoid type gear oils, they are typically some form of sulfur based EP agents that can cause degradation of yellow metals. This is a bit more of a historical problem, the newer EP additives are inactivated and require heat to become active, still best to avoid. You do not want to use a detergent oil (like engine car oils) in a machine that does not have a pressure lube system with a filter. On viscosity, ISO is more the standard, SAE can be confusing between hydraulic, engine and gear oils.
Probably the most important thing to maintaining the health of your machine is regular/annual oil changes, even with little use, water can accumulate in the oil which can lead to rust, in addition to accumulation of metal particles. I usually install a strong magnet in my oil drain plugs to retain the metal particles, you will be amazed at the amount of metal particles that you will find attached to them. It is usually recommended to break in a new lathe by running the spindle at a series of escalating speeds for for 20-30 minutes at each level, and likewise for the gears. Also run the machine in reverse. Change the oil after around 20 hours of use when new.