Are the GND pins of an Arduino identical to GND of its power supply?
&quot;Ground&quot; is just a lump of metal and each ground &quot;pin&quot; is connected to it. That includes the ground connection in the barrel jack. It's all one thing.
Wherever you connect your fan's ground to it's all ground.  That is, as long as all grounds external to the Arduino are connected, ultimately, to the Arudino.