I don't necessarily think that D&D characters are about their things. Their things certainly play a part in defining their capabilities, etc but the characters are the ones who wield them. In a by the book game, destroying their things may reduce the characters' capabilities more than death but that's a function of the hit point/save or die/resurrection mechanic in D&D rather than the fact that items are important. If a character "is" their +12 hackmaster then that's a consequence of the game or the player's take on the game being entirely focussed on combat power.
That said, as to the piles of magic items comment, I think there actually is a fair amount of precedent for it in fantasy literature and legend. In the Lord of the Rings, for instance, Frodo had the following items that would probably be classified as magical in D&D: his sword from the barrow downs (broken at the ford or Rivendell), Sting, the mithril coat, the phial of Galadriel, the elven cloak (with a special brooch), and Lembas. That's a fair pile of magical gear. Pippin had his sword from the barrow, his elven cloak (with a special brooch), some lembas, the lasting effects of the waters of the Ents, and the helmet, armor, and shield of the guard of Gondor (and, given the description, they'd probably all be magic in D&D with the possible exception of the brooch).
In Beowulf, Beowulf used several swords (and broke most of them) all of which had their own names and histories and reputations for special virtue. If I recall correctly, his shield, dagger, and mail coat had their own histories too and might well be classified as magical in D&D.
Achilles was a walking pile of magic himself. Not only was he dipped in the River Styx, his armor, shield, helmet, etc all had their own histories and special tales of their forging. (And, IIRC, Ajax and Ullysses had a rather nasty spat about who got to keep his armor after he died--now isn't that classic D&D right there?)
In the Bible, the size, weight, and material of Goliath's sword, spear, helmet, shield, and armor are all described--with corresponding implications about their expected effectiveness. While they were not magical IRL, it would be entirely reasonable for a Testament style translation of Goliath into D&D terms to give them magical properties. (And, translating the text into D&D terms, it wouldn't be distorting the message of the passage too much to say that the favor of GOD makes more of a difference to the outcome of a fight than +5 scale mail, a +3 tower shield, a helm of heavy fortification, a Large +5 longspear of wounding (monkey gripped, of course), a +3 keen large shortsword, and the fact that his player bamboozled the DM into letting him play a half-ogre with a base strength of 18--Goliath and his fearsome reputation was as much about his stats and equipment as any D&D character).
Now, when we think of Frodo, Pippin, Beowulf, Achilles, or Goliath, we don't reduce them to their things do we? Similarly, D&D characters, despite the piles of magic items they end up using need not be reduced to their things either. Even if their capabilities are heavily influenced by their possessions, that doesn't mean that their possessions define them.