Yes, mostly.
Martial abilities are somewhat abstract, but a Battle Master knows that his ability to trip someone is superior to others because of his extensive training (trip maneuver). A Champion knows that his training (or fighting spirit, etc) makes him “better” at making use of weapons than others (improved critical). A Rogue is specially trained to target weak spots to devastating effect (sneak attack). Because certain abilities tend to be grouped together, these are noticeable to those familiar with such things, and adventuring types can be classified by them. The names applied to such classes and subclasses can of course vary.
With magic and supernatural abilities it is even more overt. The fluff in the PHB is assumed in my world (unless specifically over-ridden). Wizardry, Sorcery, spell levels and spell slots are all actual things, as is the ki that empowers a Budoka (the term “monk” is used as it is in the real world to refer to monastics in various cultures, not to refer only to ascetic ki-focused martial artists), the divine powers possessed by paladins, clerics, druids, and rangers, etc. All of this is in game and forms a part of the character’s identity.
I also have a clear divide between magic/supernatural things and non-magical things that bounds fluff. A Budoka’s Ki is supernatural, for instance, and it isn’t possible to refluff the class as a brawler. Rangers are supernaturally empowered spellcasters, not wilderness warriors with some inexplicable tricks.
Part of my enjoyment of D&D involves experiencing D&Disms, and the in-world reality of such things is a D&Dism that was presented to me in almost all of my formative D&D experience.