Hmm, I like that, it does give a stronger diegetic identity.
Is class something that's detectable within the fiction, or is it more something that's demonstrated by ability? ("That guy can cast fire bolt and shatter, he must be a wizard.")
Yes. Both.
In the past, I tended to define a culture by means of which classes were prominent in it. Example, Lolth Drow were mainly charismatic Paladins and Warlocks, female academies and male academies, respectively. At this point, there was a one-to-one between specific classes and cultural narratives. The inworld might have alternate names, but the class features were explicit, and recognizable.
For 2024, the species dependence on a specific ability will be gone. Instead, each culture has its own sample of prominent backgrounds that might promote any of the six abilities. The backgrounds may relate combat training (such as a specific kind of Soldier or Veteran, or so on), but most backgrounds are noncombat professions and interesting origins. A background can be specific to a culture, such as High Elf Griffon Rider, who utilizes and cares for Griffons, for combat or noncombat.
Most cultures have some kind of combat training, whether the warriors of an extended family or a town militia, or a national army. These combat institutions will continue to cohere with actual D&D class mechanics. Again for the sake of example, the High culture Elves typically have two military institutions depending on individual aptitute, Eldritch Knight and Bladesinger. In some towns, military special units are also known, such as Moon Druid or Evoker Wizard or Diviner Wizard, but these tend to be part of the Bladesinger academy.
Assuming class identity is primarily identified by ability, how do non-magic classes assert their class? Is it able to be recognized?
In all my worlds, the martial power source is nonnative to the Feywild. If a Fey culture has any form of Fighter or Rogue, they necessarily learned it from the Material Plane, almost always from a Human culture. The High culture of Elves are competent at Fighter (Eldritch Knight) because of living nearby Human cultures for millennia. There is no such thing as an Eladrin culture Fighter institution. Any Eladrin Elf Fighter learned it from somewhere else, often from a High culture.