Dave's video is about measurement and connection to other electrical appliances, and what can happen.
When Rigol DS2072 is not grounded, it have 115 Volt (or 230 Volt) on chassis and also on probe tip.
Completely without being connected to other equipment.
All my old analog oscilloscopes I have had, do not have 230 V out on Chassis ground, and also on probe tip. And all other electrical instruments I have do not have it.
Rigol DS2072 must be bad isolated from main power(230V).
Are other digital oscilloscope brands also poorly insulated ?
Of course, it is best to ground.
So I just did a few checks. From measureing the mains input socket with an LCR meter, there's a common mode filter with 4.7nf caps between N-G and A-G. There's no blead resistance across the caps and ground (I am not saying there should be.)
So under controlled conditions I removed a ground, powered it up, and got what you described. This is as to be expected, the 4.7nf caps are forming a voltage divider to earth at the input CM filter, and so earth is now half way between neutral and active potentials. In Oz, with a 240V active and neutral at close enough to ground, this gives ~ 120VACrms on the jacks.
Shorting this to ground gives a current of 338uArms, which if you work out 1/(2*pi*f*C) on 4.7nf, and divide 240Vrms by it, you’ll get exactly that.
It’s got nothing at all to do with insulation, it’s designed that way to remove mains line noise (and vice versa.) So just operate it with a ground and you’ll be fine.