I need some help understanding the schematics of DIY HS102 oscilloscope

I'm guessing that the intent is to create an offset ground reference for the probes so that the one-sided (0 - 3.3V) measurement of the A-D can accommodate the +/-5V input claimed. The resistive dividers reduce that input down to +/-1.6V relative to the center point of the supply.

The setup of the firmware includes a step to ground the probes and calibrate the zero level, which would then subtract that offset from the input measurement, allowing a proper bipolar representation of the input voltage.

Obviously the regulator isn't configured correctly to do that - there are a couple of resistors missing from the adjustment and the pinout is incorrect. It could be replaced by another 1k resistor and leave the circuit to function well enough with just a resistive divider, but there'd be a risk of the "Probe GND" suffering a small offset from a DC input on the probes, the regulator if it were functioning would minimize that.

I agree that the 1.2V regulator's output is not used, even on the drawing of the prototype board there's nothing connected to pin 2 (which is the output).

So the LDO only loads the 3.3 V line with it's quiescent current consumption.

I also see no reason why it is there. Removing it should not change anything.

Perhaps the author first thought he needed a 1.2 V and then later discovered that he didn't. Or indeed as Marcus suggests in the comments: the author didn't have a clue and just copied the design including superfluous components.