The other part is the weapons. Think of all the heroes, in any (fiction) genre, in any medium. Most of these heroes will have items attached to them - usually one or two. Whether it is a lightsaber, Anduril, or even a pen or a voice, they have one main tool associated with them. Even video game heroes such as those mentioned in the article, like Link or Master Chief, usually have their primary, identifying weapon, and sometimes armor or other secondary item, even though oftentimes it isn't really the main or even the most common weapon you use in the game.
More games need these. Specifically, more games need these to truly be the defining weapon, and have differing uses for the one weapon instead of relying on huge inventories to provide interesting gameplay. Surprisingly, many RPGs manage not to do this, despite the fact that they are the ones that really should the best. They don't even have one at all, most of the time, instead letting you choose your own - but then they don't capitalize on it. These weapons, instead of becoming your own Excalibur, become "Uber Sword of Awesome", with no personality, no story, and no individuality, identical to thousands or millions of others across the world. The main thing they need, even if you choose it yourself, is permanence, a name, and a distinctive style of some sort.
P.S: My favorite example of a truly defining weapon, both in the minds of the fans and in the canon of the game itself: the Keyblades.
You know, I remember way back in my 2nd edition D&D games a Dragon magazine article with the same complaint: "Your fighter PC wears his Ring of the Ram, puts on his Plate Mail of Etherealness, his Helm of Brilliance, and straps on his Lion's Shield, then hefts...his Longsword +1."
It's kind of fascinating that in the mythic epics and fantasy novels that inspired D&D, you'll see maybe half a dozen magic items tops, but they're all unique and powerful, with histories almost as colourful as their owners. Even genres where the characters use guns tend to have the heroes and villains packing preferred weapons. Meanwhile, D&D treated magic items like trading your iPod for the newer, better model. Unfortunately, it seems to be the latter sentiment that's carried over into the video game world.