Probably an over zealous owner with the grease. You also have to remember it does sit close to the exhaust system so it gets hot which may break down the grease somewhat.
No the engine cannot fill the rod change housing with engine oil.

Where the shift rod goes into the transmission is a notorious point of leaking oil. You need to replace the oil seal at that point. Better yet there is a kit available made up of an oil seal and a sleeve. The sleeve goes in first to stabilize the spigot the shift rod is attached to. When you pry out the old seal make sure there isn't a second seal further in. Before the advent of the sleeve system it was common practice to install 2 oil seals to try to stem the flow.

I just bought a short shift kit for my rod change transmission.

When I took the old shifter out I noticed that the box it sits in is full of thick black grease or oil. There is also quite a bit on the underside of the box.

Is this normal? Did someone get overzealous when greasing it at some point? Is is filling with engine oil somehow from a leak where the rod engages with the transmission?

Thanks!