Mislabelled ATmega328-PU
Maybe. It might possibly be a mis-signatured '328 instead. Without testing it's impossible to say.
I don't know. I would imagine that it's rare, but I have no personal experience with this.
One thing to try is to do something only the '328P can, i.e. disable the built-in brown-out detection. Try to toggle bits 5 (BODS) and 6 (BODSE) of MCUCR and see if the changes stick.
Another thing you can do is to try to duplicate the power usage curves given in the datasheet for the ATmega328P.
If you find that either or both of these fail, your device is a mis-signatured '328 rather than a mislabeled '328P. In either case you can safely use the device as a normal '328 as long as you compensate for the signature difference and don't need the exact power usage characteristic of the '328.