Your question appears to indicate an uncertainty about the term "symbolism," especially as it is applied to this excellent short story by Edgar Allen Poe. When we talk about symbolism in literature, we are referring to an object, event, person or animal that stands for both itself and also for something else. Consider a very common symbol in our culture: a red rose. It clearly stands for both itself on the literal level, a red rose, but also has come to symbolise love, devotion and affection.
When we think of "The Black Cat," therefore, it is difficult to see how the axe is actually a symbol of anything. Note how its use is described by the unreliable narrator who so chillingly slaughters his wife:
Uplifting an axe, and forgetting, in my wrath, the childish dread which had hitherto stayed my hand, I aimed a blow at the animal which, of course, would have proved instantly fatal had it descended as I wished.
It is difficult to see how the axe itself symbolises anything therefore. Clearly, the action of the narrator in trying to kill the black cat definitely could be said to symbolise his anger and the way that the black cat has psychologically dominated and haunted him, but the axe itself seems to be difficult to attach to a symbolic meaning.