I've gotten a bit lost about what it is the OP and subsequent discussion are arguing for...or against.
- Do [or Should?] the "class names" get used -like in conversation, amongst people- and/or have a meaning [I'm guessing, as understood in metagame] something within a setting?
My answer to that is: (Do?) Yes. (Should?) Yes. And (understood in-world as they are at-table?) I'm going to say, Yes here too.
- Do [or Should?] the "class names" mean nothing in-setting and just supply an "on paper" bunch of mechanics? For example, a Cleric or a Paladin or a Divine Sorcerer with weapon proficiencies/combat feats all being able to present a, basically, "same" character type within the game world.
My answer to that is, (Do?) Not for me, no. (Should?) Not for me, no. However, "Can they...?" Yes. Absolutely. It is part of the DNA (from 3e onward. But, really, available in certain cases through multiclassing in earlier iterations) of how the game is constructed.
I think my general point/contribution to this discussion is there isn't and, imho, really "shouldn't" be an either/or situation. The game allows for, and I think, works best and presents the most fun (fairly synonymous) when both are permitted/present within a campaign's setting.
Couple examples, from my homebrew setting/world -and yes, I would submit that individual homebrew campaigns/settings/worlds IS what D&D "is" regardless of what WotC uses as their "default/example" setting for their marketing/branding/pockets purposes-
Guy walks down the street of a rough n' tumble trading market town. Not a huge place, kinda sits on the outskirts of the kingdom, but a good sized population and a favorite place for the rural folk to come sell their wares or seek out goods/services they can't get in their farming hamlets.
He's wearing some ring mail. Heavy "traveler's" cloak, darkly colored, that's clearly seen some wear. Basic leather breeches. Looks like his fur-wrapped soft boots have seen some leagues of muddy road. He's got a bow and quiver slung over his shoulder. A longsword hangs on his left hip. The glint of a dagger's pommel catches the sun at the back of his waist as his cloak flutters in a momentary breeze. Several days of growth on his face and his shoulder length hair, both sandy brown with the tell-tale golden streaks, immediately identify him as hailing from one of the southern kingdoms. But, even behind the grit of some time on the road, there is something strong and pleasing in his countenance. Those fortunate enough to catch his gaze note the sparkling light green of his eyes.
The goodwives and merchants begin to whisper from market stall to shop to stall as he makes his way into town. Looks like he's heading for the sheriff's office.
"Who is this stranger? Looks like he could be one of them Peregriff Rangers. Thought they just wandered 'round Lindril. What's 'e doin' here?" supposes the smith's wife at the washing well in the square.
"Mmm. I don't like it. He's shifty in the eyes. Beware the shining one's from the south. Clothes in light and gold, they'll cut yer throat soon as smile." says the elderly washerwoman.
"Look at that baring. He's some noble's son for sure. Maybe some officer from Duke Denil's army come to talk about the war with Ovinan. Ole Buddy at the pub said, last week, the Count's goblin armies was headin' this way. D'ja think he'd teach me how to use that sword?" the baker's son, a young man of age eager to get out of town, mutters excitedly to his friends.
"That sure is a nice bow. Might even be elvish-make. He's undoubtedly a talented archer t'have a bow like that. Lookit his eyes...he's got elvish blood. I'd bet my hat." the fletcher mentions to a customer.
"Elvish blood, you say?" the apothecary in the shop beside the fletcher/bowyer, answers. She is a beatuy of a woman. Her skill with formulating potions and healing the locals have long fostered rumors about her acumen being magical or elvish in nature. "He could be from the south. Of course, he could a northman... Might have wandered down from the Naradun. He moves swiftly, he has a purpose. He might be a hunter of those tribes...or seeking a bounty. They day the Gorunduun are trackers beyond, even, the elves."
"They who?" replied the fletcher's customer with a raised eyebrow.
Who is he?
1) Human (or half-elf) Ranger (from somewhere in the south)?
2) Human/Half-Elf Fighter, noble background... soldier background? some kind of "military officer/commander" feat? Connected to the Duke's army? A scout? A deserter?!
3) Human/Half-Elf Fighter who's really an adventurer looking for a worthy quest?
4) Human/Half-Elf Rogue? Treasure hunter? Assassin?
5) Human/Half-Elf Barbarian (from somewhere called Gorunduun)? Bounty Hunter?
6) Gnome Illusionist who's trying to infiltrate the town, knowing they arrest gnomes on sight.
The answer, to any/all, is Yes. This guy could be any of these things. NPCs could refer to him as "Fighter" or "Ranger" or "Rogue [would probably hold off and calling him a "Thief" until he did something worthy of accusation) or "Barbarian"...all terms in the game world. OR they can talk about the "Soldier" or "Bounty Hunter" or "Scout/Tracker" or "Northman"...all completely common and usable terms in the game world, as well.
Does the class name have a meaning within the world? Absolutely. Are "all persons" called this class name a cookie-cutter individual from a single place/meaning in the world? Not at all necessarily. Could be, if the setting is set up that way. But certainly doesn't HAVE to be.
A divine sorcerer who takes some armor proficiency and the cleric to the light god of awesomeness could both be present, going about religious duties, doing their jobs, in the same Temple of Awesomeness. Or they might be members of two different sects to the Light God of Awesomeness who get along/work together often, but just keep different temples/abbeys/cloisters, what have you. OR who each think the other is "wrong/heretical" in some way and needs to be wiped out. But fluff like that, that level of lore/detail, I think is probably best kept to a limited number of classes to a specific setting. Most settings, certainly the big four "defaults" shouldn't be strapped in that way.