Regarding class...yes, some of this is semantics. We're discussing if the label used to specify what in game mechanics a character possesses has the same in world meaning, so opinions will vary.
I think sometimes those labels match, and other times they don't. Which is why I don't think the term "concrete" applies.
Right, but some of us think it's at least possible for all/most of the classes to match to at least some key aspect of character identity, whereas others don't care/care less. It's a stylistic question, which is precisely why it's so important, and why it generates such strong opinions about how it should work.
As for the question about race, I don't know if I think it's the same as class...perhaps because you can't add a new race when you gain a new level?
Yes, but it became separate (in 1e) because designers thought race was a real, natural category (in the real world), whereas class was an achieved status and had to to with personal advancement. And the separation was never entirely complete (duergar that gain racial traits up to 5th level, monsters as PCs, etc.).
My point in bringing up the question of whether race is "obviously" concrete, whereas class is just a metagame concept is, you can make precisely the same kinds of arguments about race that people here are making about class.
Let's say @
Mouseferatu wants to play Friar Tuck as a barbarian, but doesn't want the fluff that goes with being some sort of Hyperborean savage.
But let's say I want to play a goblin from the George McDonald books, or a troll from Elfquest, or something similar. I don't want to design a "monster as a PC" class. I am not a monster. I don't demand that my character regenerate. I am not especially sneaky. Most of my relatives are not evil in alignment. But I'm a miner, and in fact, mechanically, what I can do is pretty much covered by the racial category called "dwarf". But I'm not interested in having a beard, speaking "dwarvish" (whatever that is), holding grudges, or hoarding gold. Why must my DM demand that I conform to the dwarvish stereotypes when really all I'm interested in are the mechanics?
So the question is, if we can view class as being about crunch and not specific archetypes, why can't we do it with race? If we're being consistent, it shouldn't detract from the game: as long as we have a well-RP'd character concept, I should be able to exercise the power of choice so I can contribute to the collective story-telling as a full-fledged participant.
EDIT: Oh, and about being able to add a new race after gaining a level: My non-dwarf troll character who uses dwarf stats has just become 4th level. In-game, I have recently learned that my mother is an elf. So I ask my DM to let me pick up a feat that gives me fae ancestry - resistance vs. sleep and charm, no need t sleep, etc.