A character concept
We feel spell levels are completely artificial, so a player concept, but this is only so because nothing in our lives is this non-gradual. If I can run a lap around the block 92 seconds today, it might be 90 tomorrow, but not 28.
This is not so in the game. Yesterday I could not cast Fireball or Fly, but today I can, 2 actually. A week from now, I can cast 3.
People are not willingly stupid in the game, they recognize
this. This also means not only spell levels are obvious, but levels of casters too. Not so much for a Rogue, the increase of Sneak Attack or proficiency bonus is quite gradual.
If your abilities are very distinct (and some are, Wizards getting 3rd level spells on level 5, Bards getting Expertise on level 3) it is obvious.
You can say characters do not see this, but that would be unnatural for them.