Power – Feeding Negative Voltage to LM338 ADJ Pin

Because there is current flowing thru the ground wire between your PCB and the common ground point. The wire has a small resistance. The current time this resistance causes a offset voltage. For example, if there were 1 A flowing in the wire, then it would only need to have 23 mΩ resistance to exhibit the 23 mV offset.

Another more subtle possibility is that you have high frequencies in the system that are confusing the meter electronics. Above some frequency, probably somewhere in the 10s of kHz to 100s of kHz range, some meters without adequate filtering may start seeing AC as DC. Old purely mechanical meters are pretty much immune to this, but modern meters with active electronic front ends provide various ways for high frequency AC to be treated non-symmetrically and therefore to add a offset to the DC measurement. That all said, this is rather less likely than the much simpler explanation of a offset voltage accross the ground wire due its resistance times current thru it.

Since you are using the same power supply for both, the Arduino and the LEDs, you do not need to connect the ground pins -- as long as ground is connected internally.

As you have recognized this would be redundant and unnecessary and could even lead to further problems (there would be several ground loops). So I think connecting the Arduino to ground just one time will be fine.