Ok, dual wielding was a thing. There are medieval fencing treatises that cover it, albeit briefly. And by "dual-wielding" I mean using a full sized 1 handed weapon in each hand, usually a pair of matched swords.
It was extremely rare though for 3 reasons:
1. It was very difficult to learn. You had to be nearly completely ambidextrous to make it worthwhile, otherwise you would end up only using one of the weapons effectively. And then it was a whole other system of techniques in addition to single-sword, sword and buckler, sword and dagger, sword and shield, etc . . .
2. It was inconvenient. Swords were popular because they were easy to carry, just strap the sword to a belt and you're good to go. Wearing two swords kind of nullified that though. The same would of course apply even more so with other weapon like axes, which are more cumbersome/inconvenient to carry. A back-up dagger or small shield/buckler was much easier to manage along with your sword than another sword.
3. It wasn't that much better than other options. The advantages to be gained from wielding two weapons at once aren't that great. First of all, it does NOT double your offensive output. Real life is not D&D and people don't take turns swinging at each other in a real fight. You can swing 1 sword with one hand just as fast as you can 1 sword in each hand. The advantage of two swords was that you could threaten two targets at once, and thus force an opponent to cover more of their body than they would otherwise have to. So in a street fight/duel situation against an opponent with just one sword, dual-wielding would confer an advantage, if you were very highly trained, and bothered to carry two swords around everywhere you went. In a battle, or against someone with a shield and sword, all of your advantages become liabilities. A shield id better at defending than a sword, and when you have to face more than one opponent, missile weapons, and pole-arms you want that superior defense.