If the Master Sword wasn't iconic enough, there is still the Buster Sword from Final Fantasy VII. By solely mentioning the name, a picture of Cloud (or Angeal, or Zack, depending on what games you've played) wielding that behemoth of a weapon pops into your head. Because of this sword, Cloud is considered physically strong, and the sword helps add to the "tough-guy" attitude he has. The sword is huge, so it sticks out like a sore thumb, and while their are other video game characters that fight with a greatsword, such as Ike (Fire Emblem), Sigfried (Soul Calibur), etc., the Buster Sword has a very distinct appearance. Unlike many greatswords, Cloud's is only sharp on one side, it also has a very broad base for the blade, and remains broad until it reaches the tip. Cloud's Buster Sword (top) looks more like a sharpened chunk of steel with holes than it does a sword (bottom).
Even in shooting games like Call of Duty, a character's presence can easily be felt by the appearance of a specific weapon. If only 1 player has a weapon with Gold Camo, everyone can recognize that player. Whether his name may be Xx420BlazeItSwagxX or ted42, everyone will know who who that player is without have to look at the name above the generic character. The weapon speaks louder than the character, and any time that weapon is seen, a player immediately can tell who is there. Other games like this include World of Warcraft and Guild Wars. Even though the characters in those games are built to the player's interest, the appearance of a unique weapon can say more than the character's physical appearance can.
However, not all weapons can point to a character. In games where party members can have any item at any time, it is hard to tell what belongs to whom. An example of this would be held items in Pokemon. While one can argue that these items aren't weapons, they aid the Pokemon in battle, much like a weapon. If I gave you a list of items, it would be impossible to tell which of the 719 Pokemon owned it. While it would make sense to have Lucario hold Lucarionite, a Lucario can also hold an Oran Berry, a silk scarf, etc., just like Hitmonchan could hold the Lucarionite.Again, the argument could be made that held items shouldn't count as weapons, but in that case, the "weapons" would be left in the Pokemon's attacks. Even on this level, there really isn't a lot of presence from an attack. While I agree seeing Flamethrower means the opponent has a strong attack, simply seeing the animation for the attack won't tell me who the user is. While the attack can't be used by all Pokemon, a large pool of them can, including Fire (obviously),Water, Steel, Fairy, Normal, Fighting, and other types of Pokemon.
In short, a character's weapon can really emphasize that character's appearance, but only if that character is the only one with such a weapon. When multiple characters can have the same epic-awesome-super-powerful weapon as you, it is no longer "your weapon".