Fury requires that you deal enough damage to force Levin Indok (the gold colored boss) to summon Levin Undok (the silver colored boss) to the fight without changing forms by switching her heads.
I found that this could not unlock by ignoring Indok and only damaging Undok after an initial head change, so I'm currently convinced this must be done before any change occurs.
To do this, make use of your parries and your spears to ensure that Indok has few chances to get away from you, so you can maximize your damage to her. She will almost always begin the fight by throwing a boomerang axe at you; parry this if you can, or avoid if entirely otherwise. If you're not close to Indok at this point, she may summon three floating daggers to take flying stabs at you. Knock these out of the air immediately by locking on to Indok and throwing your spear as the daggers come to rest beside Indok. If it's timed right, you'll hit all three at once and stop them from attacking you. From this point on, do your best to stay close and deal as much damage as possible, using your parries on her sword strikes to allow for an easy couple of hits before she attacks again. Eventually, she'll begin to throw axes either directly at you (three at once, or multiple in random patterns) or circling herself. Avoid this as you're able, and heal as you need to.
If you're able to take Indok's health to between 30% - 50% before she begins switching her head to turn in to Undok, she will start summoning Undok to the arena instead, and Fury should pop up shortly afterwards.
If anyone has any other tips they feel should be written here, or found a different method that unlocked this for themselves, please let us know in the comments to ensure the solution and discussion around it is as accurate as possible!