Please take my answer with a pinch of salt, as I’m not a trained engineer. I’m happy to be corrected by someone more knowledgeable. But here’s my understanding:
In audio electronics, Earth and Ground are not the same thing. Earth is an electrical safety system, and Ground is a voltage reference.
“Earth” is a physical point, literally a piece of metal stuck into the earth, that things are connected to, to provide safety in the event of a mains electricity short. “Ground” is a point in a circuit to which other voltages are referred to. It would be better named “Common” or “Reference”. In the case of the Digitone, it’s probably the negative side of the 12V DC power supply.
The Digitone is powered by a 2-wire supply, so it’s not Earthed. If you connect it to a piece of earthed audio equipment, eg an amplifier, it might become Earthed that way.
You ask whether ground is “running through” the chassis, which doesn’t really make sense. Voltages don’t run anywhere, it’s electrical current that flows.
The Digitone chassis may be connected to ground, this could be checked easily with a multimeter. I’m curious why you want to know this? The 12C DC power supply is not something to worry about. The safety instructions at the start of the manual tell you all you need to know.