The form list Max is referring to is called woodChoppingAxes, and can be found by typing part of that into the search box. In the vanilla game (with Skyrim and Update active) it contains two axes already, dunHaltedStreamPoachersAxe and Axe01, the latter of which is your typical wood axe.
Open the list, find your axe in the ck, and drag/drop it in. Note, this does not only take axes. For example, if you created a generic "tool kit," and added it to both this form list and mineOreToolsList, having the tool kit in your possession would cause you to produce an axe at the chopping block and a pickaxe at ore veins.
I don't know if your axe looks any different than the normal ones, but you will still produce a standard wood axe unless you also change the one associated with the chopping block. Unfortunately, I do not know how to make it recognize which axe you may be carrying, and I struck out at attempting to figure it out by looking at the ancient nordic pickaxe.