If you can equip it you can swing it. You just might not have much reason to.
Still, if you were to focus on non-shot abilities, you could probably make a melee DH work. Damage would come more out of Impale and traps, I suppose, with some acrobatics to navigate around.
As for why so many weapon types, to differentiate stats. Instead of having 50 different kinds of "sword" with varying damage ranges and swing speeds, you get the "sword" category which is always from x-4 to x+8 damage at 1.40 swing speed, the "axe" category that are x-3 to x+10 damage at 1.50 swing speed, and so on (numbers made up, of course).
At least that's the gameplay justification. Really they're just there so the game looks like it has more loot for you to find, since Diablo games are 100% focused on hack-slash-loot farming mechanics and you'd get bored if all you ever saw was "sword" "rusty sword" "not-rusty sword" "sword again" "sword" "another sword" "sword" "sword" "sword" "sword" "sword."
Looks a lot more interesting if that list is "sword" "sabre" "mace" "club" "axe" "broadsword" "cleaver" "dirk" "flail," right? But it means pretty much the same thing in the end.
Related: I was surprised to discover monks can equip most of the weapons. My monk was wielding a rare saber with an offhand axe by the time I stopped playing. Sorta expected they'd be restricted to blunt weapons, as is the trend in most games with a monk class. A welcome surprise, though.