I think Susie’s axe might be a toothbrush she carries around.

Like it’s description says “it’s formed from the mane of a dragon whelp” maybe it’s like some old kid’s tooth brush.

It would also work with her trying to avoid going home with her staying out more and brushing her teeth in anyplace she can find a sink. (Witch is not really working well for her)

And also how she comments on how she too good for it when you replace it.

Thought I could be wrong