Just because you’re not obligated to kill the monsters doesn’t mean Doom isn’t about the monsters. The whole point of Doom is that you’re running through a facility and later Hell, both of which are swarming with enemies. You get to the next level to see what else the game is going to throw at you.
And Super Mario is more than just “run right,” the point is to save the Princess of the Mushroom people (original game manual PDF). But even that backstory doesn’t make Mario an RPG, because the point of the game isn’t Mario’s story, but the enemies and worlds you go through along the way.
Your mission is to shoot your way through a monster-invested holocaust. Living to tell the tale, if possible.
There’s a backstory, but it’s pretty short and doesn’t really give you a real reason to engage with the character. You’re there to get revenge on the monsters who killed your platoon, and that’s about it.
And let’s look at Akalabeth. There’s a backstory, but more importantly there are stats: dexterity, strength, wisdom, and stamina. You get a random start, and can replay the same start by using the same seed. The game is all about the character, not the enemies or the world design. Yeah, it’s a dungeon crawler with simple combat, but the whole point is the character.
Therefore Akalabeth is an RPG since that’s where the focus is. Doom and Mario aren’t, they focus on something else (in Doom, that’s monsters, and in Mario, it’s getting to the end of the level (there’s a timer and reward for getting there quickly).