'Mobiloil' would suggest it is an oil.
Probably just an engine oil, before oils became specialized.
The general rule is that if it has a drain plug, then it was intended for use with oil.
Does it have a filler point/level plug ?
The basic box looks to have a drain plug ?
https://www.gracesguide.co.uk/images/3/3c/Im19230823MC-Albion.jpgBut the kickstart version is a little less obvious. !
https://www.gracesguide.co.uk/images/4/4a/Im19200924MCT-Albion.jpgYou could test 'fill' it with oil.
And see if it all just comes back out on the floor !
Some folks mix a little flowable grease in with the oil, to slow the rate of exit.
About every ride on lawnmower uses flowable grease in the gearbox - is it 00 grade /thixotropic grease ?
Can't help with the kickstart side of things.
If you turn it over in either of those 2 states, does it sound like a buzz saw on steroids ?
Most boxes have those separated, and the kickstart action pushes/engages them together.
Hmmm, Max 2 hp.
Sounds like a fun little beast.
Have fun !