Can a multimeter measuring current damage low voltage devices?
If your DMM is of auto-ranging type, never use it for measurement of current in a low-voltage environment. The auto-ranging DMM will change the measuring resistor (shunt) value in-flight, which will add a varying voltage drop affecting your circuit power supply.

Instead, use a 1% 0.1 Ohm resistor in-line with your power rail (between the regulator and Arduino) as a starting value, and use the milli-voltmeter mode of your DMM to measure the current, I (mA) = V(mV)*10 for a 0.1 Ohm resistor.