8x8x8 multiplexed LED cube with an Arduino Mega 2560

Un4Seen:

Why don't you take a AC voltage reading between your iron tip and an independent path to your power ground when the iron is turned on

In other words take an AC voltage reading with a multimeter between the iron's tip and the grounding in the wall socket?

Yes
The iron doesn't need to be turned on the light the LEDs when touching them. I just needs to be plugged in.

Turn the iron on for the test, but no LEDs required, just measure for AC voltage from tip to ground while powered up. Obviously you can't leave your meter lead on the hot iron for too long.

I wonder if I can demand changing it in the store because of this... I mean, is this a defect or it's just a poor model?

Make the test first. I searched a little for this product description but can't find anywhere where it states it's a grounded iron tip or that it offers ESD protection which is the same thing.
Lefty