Is there a hidden axe location?
The developer says an axe is not normally obtainable in the demo. There is no normal pickup route to follow for the demo version described in that answer.
The browser Controls screen lists F as Chop. A key binding can exist even when the corresponding item is unavailable, so pressing F is not a way to obtain an axe.
Why videos show an axe
AiminXP’s “Trees Fear You…” video uses a mod by Fockster. Its description and the mod creator’s post identify that connection; the mod adds an axe-related action.
The developer also describes hidden axe code being used by mods. That does not turn a modded recording into a normal demo walkthrough.
What about the full game?
Do not treat the demo’s item availability as a final-game item list. This answer covers the demo, not a promised axe location, unlock condition or mod compatibility for the full release.
If you are looking for a collectible you can keep, start with the Top Hat.