The character absolutely should have an aspect about the item of power. But I agree that Bearer of Bjornedyr is a bit boring. Think about how he feels about the item and what it represents to him. That should lead you to what sort of invokes and compels you want and that should help form your aspect. Here are some examples.
Honored by Odin with Bjornedyr - If the character likes Odin and getting the axe was the greatest moment of his life. Invoke to be in Odin's favor, compel to have the character need to stay in Odin's good graces.
Exit Through the Bear Door - If the character uses the axe to escape from situations. Invoke to be able to leave a situation, compel to force you to leave via the nevernever even if it isn't necessary.
An Axe so sharp it can cut Reality - If the character takes a more secular view of the axe. Invoke to chop things, compel to have the axe accidentally cut something.
The Onus of Bjornedyr's use - If the character sees the axe as a responsibility. Invoke to stay true to Bjornedyr's purpose, compel to have the character take responsibility for more things (like things that escape to/from the nevernever through the holes that he cuts or for actions of previous bearers of the axe)
I wrestled a bear in the nevernever and all I got was this lousy axe - If the character isn't exactly grateful for the axe. Invoke on Wrestling, compel for the character to express ingratitude.
Now I kinda want to use that last one. I went into a frenzy in Norway and all I got was this lousy shirt or something like that for someone who became a berserker (literally bear shirt).