Why does my ungrounded DC power supply work?
The lab power supply has three banana sockets, red supply output, black supply return, and green mains earth.

When you have the metal bridge in place, it connects mains earth to black supply return. Since your PC that powers the Arduino has a metal case that is also mains earthed, it means that all the return currents from the 9V lab supply returned via motor driver to Arduino, and from there to PC via USB ground, and from PC back to lab supply black terminal via earth wiring in the house mains wiring.

It is just hard to see as you did not draw the PC in there, or mains earth wiring between PC and lab supply.

That is exactly why the grounding should be considered extremely carefully in a system. Fortunately, most likely nothing got damaged yet, but imagine if the lab PSU currents had been larger.

So take the metal tab off so it does not short the green mains earth and the black return terminals. Sometimes this kind of floating power supply setup is extremely convenient.