I'm just brainstorming out loud so feel free to ignore this post...
Round like an apple = Hmm, probably not symmetrical or "perfect" roundness
Deep like a cup = Hmmm, deep...
yet all the king's horses can't pull it up! = Totally up in the air here, but I'm going to say that its probably metaphorically impossible to pull it up, so its either A) impossible to pull up because its not an actual object or B) something about the object makes it impossible to do anything with
****
Reaching those thoughts and conclusions, I'm going to guess ... a hole??