And is there any part of this statement that is rendered any less true if a character can possess multiple classes?
It's assuming characters can do that. But, such is only optional in 5e, so when that option is exercised, the statement is "more true" in the sense that exclusivity is more pronounced and applies to more featuers.
Respectfully, I am not sure where the distaste for multiclassing started!
In 1e demihumans were able to do so. It was assumed that they had more than one sort of knowledge/ability baked into their 'concept' and no one I knew ever balked at it.
A lot of people balked at 1e MCing. Not that demi-humans could do it, but that they could do it with only certain class combos, and had level limits, and/or that humans couldn't MC.
In one of the basic sets (not the one I played c1979), there wasn't demi-human MCing, instead 'Elf' was essentially a Fighter/Magic-user class in itself - kinda like a 5e EK or Bladesinger. IDK when MCing made it into 0e.
Now, some express concerns about multiclassing being too artificial. Perhaps that is because you can "suddenly" take on a class which seems to have no relationship to your history or character. I guess I can see where there is an issue for some campaigns here.
And not an issue for 1e non-/demi- human MCing, which was MC'd from 1st level on. It's long been an objection to 3e-style 'modular' MCing.
What does blow me away is the absolute hatred for multiclassing as artificial or contrived. The whole concept of classes and levels is artificial and contrived but we are conditioned to accept them.
I thought we'd long since been conditioned to accept multi-classing, too.
If you do not think it fits the lore, I might ask in which case and in which what world. Because hopefully you are not telling me you cannot fathom a fighter/rogue...
Classes are innately limiting, and being able to mix them opens up the possibilty of modeling more concepts without adding too many more classes... In an old Dragon mag, there was a " 'Bandit' (Unofficial NPC) Class," that was essentially a Fighter/Thief, so a human could mix the skills of both (I played one through 8th level back in the day, fairly awesome, in large part because of magic items, and a little broken, because whoever designed Bandit kinda 'averaged' the two classes, so it advanced /faster/ than a fighter). That's an example of what more open MCing gains you (helps you avoid).
Taken to the logical extreme, a 'pure' class-based system would need one class for each and every character concept out there.