This won't be a conclusive answer, but...
...I don't see a huge contradiction that can't be explained away. Cave paintings in real life are not exactly equal to the subjects they portray:
Even within the context of this TV show, the paintings done by the Children of the Forest are not particularly realistic (I think Neanderthals actually had better artistic skills!):
As for the picture of the Night King, if you squint your eyes it kinda looks like him; all major features are there. If you squint your eyes even harder, the "beard" could be conflated with the thingy around his neck. With some artistic license, sure, but it doesn't seem like an outrageous mistake to me.
As an aside, note that the "humans" in the Children of the Forest drawing don't look like real humans either, but this can't be called a "mistake" or continuity error; it's just that the Children weren't going for a realistic arts style, just for a non-realistic but figurative one. Or maybe they just weren't very good artists ;)