Can I attach Spinning Axe to my unit after my unit dies even though it's not my turn?

No — you can't re-attach Spinning Axe on a turn that isn't yours.

[Equip] is an activated ability, and activated abilities can only be used on your own turn, in an Open State (chain empty, no Showdown) — Spinning Axe's Equip doesn't have Reaction, so it doesn't get special timing. Also note the Axe doesn't return to your Base immediately when the unit dies: it detaches and stays at the battlefield until the next Cleanup Recalls it. So on your turn, once the chain is empty and the Axe is at your Base (after that Cleanup), you can pay its [A] Equip cost to attach it to another unit you control.

Rule 381 (activated abilities on your turn, Open State only) and Rule 457.1 (unattached gear recalled next Cleanup).