Nethack/WMG

The Wizard of Yendor made the Amulet of Yendor into his Soul Jar.
It seemed like a good idea at the time. It's normally indestructible; so there's no earthly danger and he can constantly come back to life as much as he wants. Taking it into the Astral Plane and beyond, though, would snap the astral line connecting it to him; so this is why he doesn't care if the High Priest of Moloch has it, but goes after you.
 * This makes a surprising amount of sense (among other things, it explains why he's called the Wizard of Yendor instead of the Wizard of Egypt).

The game is one large Extreme Graphical Representation showing a hacker's exploits to find a valuable file.
This not only provides some integrity to the name, but it also explains the Anachronism Stew, particularly the grid bugs.

Pestilence, Famine, and Death are actively trying to kill you (even before you meet them in person).
Pestilence is the reason it's so easy to get poisoned from a corpse that should be perfectly safe, or no match for your mighty (example: Orcish) immune system. It introduces bugs that are invincible, not normally around you, or both. Famine makes the effects of, well, famine happen to you more quickly. Death has a more difficult time of it, considering most of the ways you can die are covered by Pestilence, Famine, or War, but it's willing to take you as soon as it gets an opening, and will even take you out of turn for injuries that directly result from war despite. The reason they take so long to show up in person is that most of the Horsemen aren't used to having a corporeal form and it's difficult to get all the metaphorical (or metaphysical) paperwork filed any more quickly.

All of the games by a given player take place in roughly the same continuity, and all of the players are reincarnations of one another.
You're playing a game against the remaining horsemen of the apocalypse, and the god/ubergods/ allow you to respawn every time you die.

You are not War. At first.
The whole point of this endeavor is that you will get the Amulet to the celestial realm and hand it over to your God, who will reward you with immortality and minor godhood. And when you start, you are mere mortal, with nothing special about you. You are not War. You are not War's avatar. You are just a very young adventurer, diving into the Dungeon's of Doom on a quest.

But during the game, you or your pets will murder a million creatures, and this will often involve genocides (Magical or mundane). Killing so many things over such a short period naturally will get the attention of cosmic forces, above even the gods. You are not War, but after being directly or indirectly involved in such a ridiculous amount of combat in such a short amount of time, you will have become War. You replaced the previous War by being so much better at it than him. You murdered your way to the bottom of Gehennom, ganked the Amulet, and then carved a path back to the surface and into Heaven. You're badass incarnate, one way or another. You're War now. Or almost.

If you die at this point, you stay dead. The Horsemen very much do not, and will annoy the crap out of you by getting up after being rightfully slain unless you get tricky. You're clearly not QUITE War yet, but if you got that far and have that much power, you're clearly intended as his successor. This is the reward the gods give you; you are War in all but an official sense, so they make it official.

Death's taunt ("Who you you think you are, War?") can support this. It could be an insult. It doesn't mean "War, who do you think you are?" but "So you think you're War, huh?" and is intended to be dismissive. He's trying to kill you and is being kind of a dick, not reminding you that you're a Horseman (Because you're not, yet).