The Fighter is, and kind of has been since 2e-ish, a catch 22.
They are the character with the least amount of "built-in flavor"...They are the class that fights. It's what they do. They're the best at it.
Sword and shield? Mounted knight in shining armor (when you didn't want the "holy baggage" of the paladin)? Throwing axes in studded leather woodsman (when you didn't want the "Aragorn-baggage" of a ranger)? Chain wearing battle-axe wielding madman? Swashbuckler/buccaneer/pirate? Cavalier/chevalier/knight (before and even after there was a Cavalier)? Mercenary/solider/guardsman? Amazon/Valkyrie...Xena? Samurai? WWF-inspired wrestler/brawler? Kid with a borrowed (or stolen) sword who'd never left the farm before? Hercules and Perseus. Conan and Lancelot. Blackbeard and Gilgamesh.
The Fighter was, purposefully, left to handle all of these (and many more) archetypes. And, through the majority of the editions (and iterations of fantasy TTRPG clones), this is what they have been. The Fighter has been a class of its mechanics devoid of any true single core "identity" of flavor and story to fall back upon.
5e is no different. They built the Fighter and their subclasses around mechanics, not story. The "simple" fighter. The "slightly more complex/mechanically interesting" fighter. And the "slightly more complex than that because it involves magic (spell progression, spells known, and all that)" fighter.
Is an Eldritch Knight a literal "knight" in plate mail that belongs to a sanctioned (or unsanctioned) order of arcane adepts? Is it just an elfin warrior-mage trained in the skills of both because that's what elves (or elves of a certain social standing) can do? Is it a tribal warrior who has learned just enough of this shaman's teachings to generate magical effects to assist his fighting? Or Elric of Melnibone mixing innate ancestral magical ability with swordplay to conquer the known world?
The answer is "Yes." It is all of those things. On purpose. By design. As is the Battlemaster. As is the Champion. The fighter is the class in which you get to make the MOST story for your character, because other than "the Fighter fights [with weapons]" the class is given (and has) no single story of its own.
So, you come to the issue of, "The Fighter has no identity. It doesn't have any interesting/fun mechanics like paladins or druids or warlocks get. It's just boring."
When, at its core, the Fighter's "lack" of identity IS its identity.
The Fighter fights. That is the extent of their story and, simultaneously, the root mechanical framework upon which you can paint your greatest heroes and villains as you see fit.
If the Fighter "needs" anything, and I do not ascribe that it does, it would be a "fully/very complex" subclass option, with multiple moving parts and player choice options (a la a warlock), that are not "magic" upon which people that prefer a "later editions style of play" can have their "superheroic -but its not with magic- warrior." That would fill the Fighter class, as far as subclass options. Simple. Simple +1 special mechanic. Simple +Magic. Simple +2 (or more) special mechanics (a.k.a. "Not Simple").
It is a curiosity, that of all of the classes that have been developed over the years, all of the nooks and crannies of mythologies and legends and histories and cultures from which D&D has drawn PC classes...the Fighter, a warrior guy who can swing a sword, still is just the tabula rasa class that you can style however you like. No other class has this distinction.
Starting, arguably, from the Magic-user or at least from the original Thief/Rogue concept on down, every class that had come after, the built-in fluff/story of a class have become more and more specific and narrow. Some open up and some narrow further, here and there, from edition to edition. But always, there is some degree of an assumed (and generally accepted) backstory to go along with the class.
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