Right, but that's exactly what I mean: that's taking the game mechanics and abstracting them into an in-universe narrative. The hypothetical characters in that scenario aren't talking about taking levels in Fighter or getting an Extra Attack: they're using generic terms that we understand don't directly map onto the procedure of rolling particular dice.
By the same argument, how does a character distinguish in-universe between something they could learn X months down the line adventuring and something an NPC just did that they can't? From the character's perspective, both of these things (say, the PHB spell fireball vs. 'Fiery Blast (Recharge 5-6)' on some villain's statblock) are currently unknown abilities. They don't know - or shouldn't know, I would argue - that one is available to them as they gain experience and one isn't.
And even as a game structure, tbh, how are players even picking up on the fact that Fiery Blast (or whatever) is more damaging than a regular fireball? Or that it recharges rather than uses a spell slot? Or that the gladiator is rolling extra dice for damage without a feat that's in their book? Just...don't tell them.