The axe and trident should be able to be built as a proper weapon using the anvil.
Axes can already be enchanted with Sharpness, Smite, and Bane of Arthropods, and are sold by Weapon Smiths, so they are obviously intended to be used as weapons. However, they cannot be enchanted with Looting, Fire Aspect, or Knockback. Personally, I can't consider it for weapon status without Looting. Fire Aspect and Knockback are also weapon enchantments that should be usable by an axe. The only weapon enchantment that doesn't make sense is Sweeping Edge so that one should not be available.
The trident can only be enchanted with its specialty enchantments, Mending, Unbreaking, and Curse of Vanishing. Looting and Knockback do not conflict with anything, and so should be available through the anvil at least. Fire Aspect doesn't make sense for the trident, so it should not be available, and neither does Sweeping Edge.
As MacchuPicchu suggested, the most logical way of handling the other damage increasing enchantments would be to have sharpness available for the trident, but BoA and Smite should not. (Although an argument could be made for Smite because of the Drowned. A Smite enchanted trident would be a Drowned specialized weapon) As such, Impaling should be treated like the other damage increasing enchantments and be mutually exclusive with Sharpness (and Smite?).
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