I agree about the unicorn. As for this one, the logic is simple.
Why not a mechanism? Because this creature wasn't created, but born.
Why an elemental? Because the creature has a direct connection to the forces of nature, without being man-made. That is, it's not a golem.
As for the element of this creature, water and earth together create wood.
It's strange, but there is logic in your reasoning!
I sincerely believe that gargoyles should be mechanisms, for the same reason.
And you are right here too; gargoyles were created, just like golems. Yes, they should be classified as mechanisms.