Not earth grounded old analog oscilloscope
The earth ground pin of the power supply is for safety only, it does not have any impact on the signal measured by your scope probe.

The scope channel input measures the voltage between the two pins of the BNC, the inner pin and the outer pin (acting as a shield). You must always connect the two to get a proper voltage, because a voltage is the difference between two electrical potentials.

You do not always want to measure the voltage between one potential and the earth ground. For example, you could observe the voltage between the ground of a digital IC, and the earth ground.