Lots to unpack here but I'd say that a Narrative 'Mechanic' is any part of the game that lets someone at the table control the flow of the story (or narrative) mostly outside(but not always) of what their actual character can do.
Fate: While Fate points are a 'meta-currency' as
@Reynard put it, I wouldn't say they are the 'mechanic'. The mechanic is Aspects. Fate Points are, often, what gives you narrative control of that aspect.
So, someone can be 'PRONE' (an aspect), it has no narrative implication unless a player or GM makes it so by invoking the aspect. They can be PRONE the entire scene but it never takes centre stage. But let's say the guy tries to attack you and the player says, He misses because I get a +2 to dodge by invoking his PRONE aspect or more cinematically, "No, I'm going to have him miss my as character because he easily jump over his weapon because the guy is prone." The PRONE aspect now has centre stage and has influenced the narrative because the player used the mechanic embedded in FATE to change the story.
FATE also gives you narrative control when you Take Out an opponent. You can dictate what happens: they surrender, they die, they go unconscious, they give you the MCguffin, they become your servant - whatever is reasonable for the fiction. There's no meta-currency here. It's just a rule of the game. Some DMs in D&D will allow players to narrate how they defeat enemies. In FATE, it's codified in the rules and, therefore, a mechanic.
D&D
I included posts that brought this to mind:
It's true that BIFTs were poorly executed in 5e and that there wasn't a lot of guidance on how to use them or Inspiration. As they stand, they CAN be used narratively, depending how your table chooses to implement them. My group allow FATE-like abilities to declare details about the scene, or the game world. At one table, the DM also required us to describe HOW the Inspiration gave us advantage before we were allowed to spend it.
In D&D, I'd say the biggest narrative mechanic is the HELP or AID action. It requires no roll and only requires you to be next to the person or enemy that you are aiding. You see it all the time with familiars:
"the hawk swoops in and flutters around the enemy's head, distracting them so I can get my sneak attack."
But it doesn't have to be fluttering. IT could be making a loud noise, pooping on the creature's head, boosting the attacking character's morale - whatever.
Most tables have a basic requirement to at least DESCRIBE how your aid action is helping. That could be kicking sand into someone's face, or flanking them or trying to trip them or shining a light on the door as the rogue picks the lock. For most tables, "I do the AID other action" isn't good enough.
And, in that sense, that mechanic is strictly an In Character action that describes the story.
Maybe some examples might help?
Is Inspiration from 5e D&D a narrative mechanic?
If so, how so?
If not, why not?
But inspiration is awarded for playing according to BIFTs, which are a part of the character. So it’s not entirely separate of the narrative. It’s about reinforcing the character… which I would say is essential to the narrative.
BIFTs are not an intrinsic part of the game, and have been dropped since 2021.
In theory, it is pretty nebulous and table dependent, a good mechanic should be clear on when it is used and they are in the DM gift, in my mind to be really useful the player should know what to do to reliably obtain them.
Once awarded they are pure metagame.
the advantage system allows players (and DM) to describe tactical actions without worrying about specific bonuses or ‘rules’, instead they can just have characters Do and the world responds accordingly.
Even the roll itself by defining two possible outcomes creates a space where narrative can be inserted if the DM/player wants (this is especially so with disdvantage if one roll succeeds and the other fails. Its not raw but it does satisfy conditions for Fail forward or succeed at a cost )