The axe in 'the man with the axe story' could hold a more symbolic significance. It might represent the man's connection to a traditional way of life. If he is a woodsman or a rural character, the axe is an essential part of his livelihood. It could also be a symbol of change. Maybe the man is on a journey of self - discovery and the axe is what he uses to break down barriers, both literal (like cutting through bushes) and metaphorical (breaking free from old habits or limitations).
The axe might be a symbol of power or danger. In the story, depending on how the man uses it, it can represent his ability to take control of his situation or it could pose a threat to others if he uses it maliciously. For example, if he is in a conflict with other characters, the axe becomes a weapon that gives him an edge.
It could be a tool for survival. If the man is in a wilderness or a difficult situation, the axe might be used to build shelter, cut firewood for warmth and cooking.