My guess is that the DC power supply is an SMPS so will have a Y2 capacitor which will leak a small current to the secondary side which is probably connected to the tattoo gun's metal frame.
This is bad design, a low leakage supply should be used, a tattoo gun is like a medical device and should have similar safeguards built-in to the design.
The tattoo guns I have seen had a diode across the coil to kill the back emf, just like on a relay coil... Try putting a 1n4007 across the terminals...