It is a simile not based upon meaning but simply upon rhyme.
Skunks have absolutely nothing to do with alcohol consumption or inebriation.
Skunk and drunk rhyme. That is the extent of the simile.
That makes it incredibly stupid.
As stupid as:
Big as a twig
As sad as glad
As nice as an improvised explosive device
Except these aren’t universally utilized similes.
Drunk as a skunk is.
Please help me put an end to it.