While that's true, it's not the whole truth and overlooks the extent to which class features do reinforce identity.
A good example is the wizard's Spellbook feature. So wizards are on the look out for other Spellbooks, maybe friendly wizards make a habit of sharing one another's spellbooks, while rival wizards try to secure their spellbooks from rivals, etc. It implies this whole scholarly sub-culture which wizards are involved with or at least a lens through which to view the D&D world.
Another good example is the rogue's Thieves' Cant features. It implies rogues have a network of shady characters and a special means of innuendo to communicate among this sub-culture without the dominant culture understanding. It implies a roguish sub-culture and offers a lens through which to view the D&D world.
And so on for clerics selecting a god, Paladin Oaths, warlock's Otherworldly Patron, Sorcerous Origin, etc.
Of course a player can craft a PC with unique backstory without any mechanics to reinforce that concept, however a player who chooses to play any other class does have some mechanics to offer a lens & sub-culture, whereas a fighter PC is in the unique position of only having the backstory the player crafts. It has no lens & no sub-culture implied by any of its mechanics, and therefor less identity built into the class than every other class.
Whether or not this is an item of concern is a very subjective matter.
But therein lies the rub.
How many threads if "fighter needs non-com-class festures" fo we have driven by how many **vs ** how many "can gods muck with clerics who go agsinst... " and "my warlock Patton is bring used H my GM to..," and Paladin oaths and yo maybe lesser extent rogues and thieves' guilds and "my hm uses my parents backstory against..." etc etc.
The classes present features that vary, I think intentionally, in the degree of "baggage" that ties them to other things in the setting and the degree to which the character is bound to other things, groups in the setting.
The fighter (depending on sub-class) weighs in at the lowest end of that scale- minimally tied by need for weapons and armor if thsts sll the player seeks to instill into the character. It's kind of similar to the class short rest focus - more independence in choices. Some barbarians likely fit that mold as do some rangers, some sorcerers and maybe some druids as well.
But each choice has consequences. Choosing the path to fewer ties also cuts out opportunities. It's the upside and downside of the homeless orphan safety net that's not rare by any means.
On the opposite side of that spectrum, you have the warlock, cleric and paladins for instance who have direct ties that almost always shape and define and in some ways limit their play going forward.
In between those two extremes are classes like wizard where interaction is needed to get the most (as you allude to) and the rogue's secret languages to the "guilds" or whatever.
So, to me, as they discussed with wanting some class and dub-classes to be simpler in play, they have a mixture of classes each with differing degrees of "external baggage" still allowing some range within each thru backgtrounds, back story, even race.
Should every fighter get festures added that leash it to former military units, former military campaigns? Should each choice of fighting style be managed in rules like the warlock patron and cleric domain are is - chaining its mechanical gains to a specific, limited set of external campaign elements that can then serve as both boon and bane at times?
Or is it a feature - both good and bad - that the fighter class was chosen to **not** make that choice of fighter style mechanics be limiting to what your charscter's ties are?
The way I see it, if you did have fighting styles wed to any limited set of external baggage, we would be seeing the uptick in "my gm wont let my fighter..." and the like complaints from those players who seek to avoid class-fixated-ties.
So, in regards to mechanical ties and baggage by class features, I am glad they provide a range of them for folks to choose from and ways to dial it up thru backgrounds and other choices rather than choose to make all classes fit the same mold.
My games session primer makes this an obvious factor of chargen- that some classes and backgrounds come with more or less built--in ties and baggage and that one should choose appropriately - specifically calling out the high end of clerics and warlocks as some of the baggage heavy. We see these as features not bugs and drawing attention to them at chargen as the GM is a way I help players get the best fit for what they want to play.
I think the game and chargen would be duller if the classes all fit the same mold in this regard so that hermit sorcerers, second-story-burglars, wandering sell-swords and holy vessels of the light of dawn all hand pretty much just the same built in and mandated class flavored degree of "external ties".
