Melee enchantments affecting damage can be added to axes using enchanted books and the anvil.
However, the other melee effects are ignored: Knockback, Fire Aspect, and Looting are not added to axes from books. All three effects work as expected on the axe after the effects have been applied in creative mode, though.
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I'd be willing to bet that Mojang's simply forgotten about this issue. From a game design perspective, it doesn't really make sense to make the use of axes as weapons so practical in singleplayer (Smite V iron & diamond axes they can now 1-hit Zombies and Zombie Pigmen without critting) and then withhold that practicality by giving swords more enchantments. It's unfortunate that farming wither skulls is going to be that much harder because of this.
My little diamond axe had so much to look forward to.
"When I grow up, I'm gonna slay wither skeletons!"
"Axes are weapons too, am I right guys? =)"
"Swords, axes, all shapes and colors, we're all equals here!"
Now his hopes are all gone, his spirit crushed.
AX LIVES MATTER!
And what am I going to do with my Looting III book?