Although the question was not aimed at me, I hope you won't mind if I provide my take on it.
I consider the fluff of the class to be up to the player (within the constraints set by the DM's opinion of what fits in the setting) based on implication from the text in the PHB (and what it doesn't say), my experience with the D&D product across multiple editions, and a comparison with other game systems. I fully acknowledge that my interpretation is not the only one: the examples below are intended to explain the source of my interpretation, not to try to prove that my interpretation is the best one.
From the PHB:
Page 11: "You also invent the personality, appearance, and backstory of your character." If the personality, appearance, and backstory described in the class fluff was meant to be a stricture, rather than a suggestion, I would expect this quote to instead say "you invent the personality, appearance, and backstory of your character within the boundaries set by your choices in the steps listed below". The lack of such limiting language along with the placement of such an expansive directive to "invent" in the very first paragraph of character creation suggest to me that the context of character creation is one of player-driven invention.
Page 11: "Or you might be more interested in an unconventional character, such as a brawny rogue who likes hand-to-hand combat...". There is no mention of brawny rogues anywhere in the rogue class fluff. Instead, the fluff explicitly says that "rogues prioritize cunning over brute strength" (PHB 94). We know that brawny rogues are explicitly allowed by the text on page 11. Since the rogue fluff text does not make allowances for brawny rogues, I conclude that it is reasonable to infer that the class fluff text is more likely to be a series of examples and possibilities rather than strictures.
Page 11: "Class broadly describes a character's vocation, what special talents he or she is most likely to employ when exploring a dungeon, fighting monsters, or engaging in a tense negotiation". This passage says nothing about class determining personality, appearance, or backstory, even though most of the classes include fluff text that describe those things. Because they are uniquely emphasized in the introduction to choosing a class, I infer that that the "special talents" in the class description are more fundamental to each class than the fluff.
Page 45: "Class shapes the way you think about the world and interact with it and your relationship with other people and power in the multiverse. A fighter, for example, might view the world in pragmatic terms of strategy and manuevering, and see herself as just a pawn in a much larger game." (Emphasis added.) The key word here is "might", which strongly implies that there are other possibilities. Also, there is no directive to see each class's entry for specifics (and the specific example given isn't found in the fighter class description). So while class is broadly relevant to how you interact with the world, the soecifics appear to have been left up to the player.
Page 46: The first paragraphs of the Barbarian class fluff describe three example barbarians, only two of which are tribal. The rest of the fluff goes on to present a very tribal-centric description, which, if the fluff is mandatory, creates a contradiction with the non-tribal dwarf character that is explicitly allowed. This suggests to me that the fluff isn't intended to be mandatory.
Page 51: The first paragraphs of the bard description describe three possibilities: "scholar, skald, or scoundrel". Yet the "Learning from Experience" section describes bards only as entertainers, going so far as to describe that they "liv[e] on the gratitute of audiences". If you interpret the fluff as rules text, would also have to be entertainers (and live on gratuities), even though that is in tension with the idea of scholar or skald bards. It seems more reasonable to me to treat the fluff text as suggestions, in which case the contradiction vanishes.
Page 82: "Whatever their origin and their mission, paladins are united by their oaths to stand against the forces of evil." I see no way that quote can be rules text, because it contradicts other rules (the subclass choices) that permit paladins that haven't sworn oaths against evil. A clear example like this of fluff text that can't plausibly be a rule I believe supports the inference that fluff text isn't a set of rules in the first place.
Page 94: (See discussion for page 11 and pages 163-164.)
Pages 105-106, 108-109: This one is independently controversial, but I would note the tension between some of the text (arguably) assuming warlocks are obligated to their patrons and the possibility of Great Old One patrons to be unaware of their own warlocks. This suggests that the fluff text is not intended to be a stricture--if it we're, presumably more emphasis would have been put on avoiding apparent contradictions.
Pages 163-164: The multiclassing optional rules dramatically expand the number of character permutations, and include explicit rules on how to merge the mechanics of each class. There is no mention whatsoever about how to merge the fluff, even though the fluff is even more of a contradiction in many cases. For example, if class fluff was a binding stricture, a multiclass barbarian/rogue would be required to prioritize cunning over brute strength (PHB 94) which interferes with taking advantage of the barbarian's strength-based class abilities. Because the multiclass rules go into detail on class mechanics, but ignore class fluff, I think it is reasonable to infer that the class fluff must not require rules to reconcile, which it would if the fluff was itself a rule.
Broadly speaking, the language of the class fluff suggests to me possibilities rather than strictures. Where multiple options are explicitly provided (e.g. scholar, skald, scoundrel for bard on PHB 51) I read the language as inclusive rather than exclusive. By contrast, the mechanics sections use more definitive language
Comparison to Recent Editions:
Particularly since recent previous editions were quite explicit about the distinction between rules and fluff, had the designers intended the fluff text to be equivalent to rules text, I would have expected them to be quite blatant about saying so. It is true that 5e lacks the explicitness of recent editions in saying that fluff and rules are distinct. But I think it is more plausible that this lack implies a desire to make it easier for DMs to decide to enforce fluff as sacrosanct at their tables, rather than an unspoken 180 degree reversal from the idea that fluff is mutable.
Comparison to Other Systems:
If the designers had intended fluff to be sacrosanct, they could have been much more explicit about limiting player choices to those presented in the fluff text. Other games do this. For example, for Urban Shadows, after choosing an archetype, you pick your character's personality from a list of three class-specific, one-word adjectives. By contrast, see the broad language in the 5e PHB above about player invention suggests much more freedom.
Based on all of the above, I infer that class fluff is not intended by the designers to be rules text, and instead merely as possibilities and suggestions. DMs are, of course free to change that at their table, and the designers made it easier to do so in this edition.
