It is a reference to A Monster Calls, which Jim Kay also worked on and illustrated. It is apparently "a sort of Green Man".
In short it is a Green Man which was a reference to Jim Kay's work on A Monster Calls. See the bottom half of my answer for an email I received from him that this is the case.
The creature on the front cover of A Monster Calls does also look a lot similar to the one on the dust cover.

I haven’t actually read or seen this either but Jim Kay also worked on the film and again the creature in the film is very similar.

I emailed Jim Kay from his contact form on Creepy Scrawlers Ltd and this is the response I got.
Hello,
I've just noticed this creature on the Dust Cover of Harry Potter and the Chamber of Secrets (Illustrated Edition).
https://i.stack.imgur.com/44eE6.jpg
I've also seen that apparently it was revealed at some point what it was (potentially in The Prisoner of Azkaban illustrated edition but I can't see if it is).
Have you revealed what it is and if so what is it?
It's a sort of Green Man, I illustrated A Monster Calls, which kind of changed my life really. Wouldn't be illustrating today if it wasn't for that book by Patrick Ness. I worked on the film too. It was just a little nod to that really.
There are other things people haven't noticed yet. On the Chocolate bar wrapper in the chapter opener inside Prisoner of Azkaban, there is a black dog hidden among the hares. Best Wishes
J
