Skills that increase melee damage will also increase the thrown buzz axe damage (notably, the Mania skills). Items like Maylay (melee "Roid" damage bonus) Shields, Strength Relics and Class Mods that increase melee damage also affect thrown axes. However, a thrown buzz axe doesn't always count as a melee attack.
In the mission, Bad Hair Day (which requires you to do a melee attack), you won't be receiving the item if you deliver the killing blow by throwing the buzz axe.
In the 'Tina Tiny Assault on Dragon Keep' DLC mission, 'A Role-Playing Game', where you are required to punch someone (deliver a melee attack), it won't register as a melee attack if you throw the buzz axe.
If you're using the Love Thumper shield (description: when shield is depleted, in addition to normal Roid bonus the shield also causes an explosive nova every time the user connects with a melee attack), you'll release a nova when throwing the buzz axe. It'll still come from your character, not the buzz axe itself.
Weapon Fire Rate boosts/bonuses from items and skills affect the thrown buzz axe. It's not treated as a melee attack in this case (which are unaffected by weapon Fire Rate boosts).
With the Taste of Blood skill (description: "During Buzz Axe Rampage, all damage you suffer is reduced. Damage reduction is incresed for each stack of Bloodlust. Killing an enemy during Buzz Axe Rampage adds extra stacks of Bloodlust."), a kill with a thrown buzz axe won't give the additional melee kill Bloodlust stacks.
If you have the Buzz Axe Bombardier skill (description: when thrown, your buzz axe now has Dynamite attached to it which explodes on impact) from the Bloodlust skill tree:
Critical hit damage
Critical hit damage affects the thrown buzz axe like on guns, not like a melee attack. In general (not considering the explosive damage bonus from the Buzz Axe Bombardier skill) you'll deal two times the damage when you crit but the 'Pain is Power' skill lowers the damage.*
* Source: Gearbox Software forums post by user Dank Rafft, with critical hit damage tests