"Side channel" information on laptop audio jack?
If you measure between the signal lines and some external ground, any signal which appears that does not show when you use the device's own ground is ground loop noise: it is the signal representing a fluctuating potential difference between the external ground and internal ground.

The currents which charge the laptop are generating some voltage through the ground connections, which means that these different grounds are not at the same potential, even if they are in fact supposed to be the same ground, electrically.

This is not a valid signal. Audio people pull their hair out eliminating unwanted ground loop noises which sometimes appear where they should not.

You're going out of your way to pick up stray signals by tracing the potential difference between some external ground and an output terminal on the device. Your measurement spans the entire device and its power supply.

The headphone jack's signals consist of the voltage between the tip or ring, and the sleeve. That ground, not any other.  If you use the wrong ground, you are not measuring the signal properly across the points between which its voltage is supposed to appear.  It is not some "side channel" data, but rather garbage.

This is like measuring someone's height in shoes, and noticing: hey, there is some "side channel" of extra height data that varies with whether shoes are worn and what type!