So in summary: you're saying you can't play a warrior without calling it a [insert warrior class name here]? In game terms that makes sense, because a class' name carries with it all the trappings of said class; and just because I want to play a Thief as if it's a musclebound warrior doesn't mean it isn't still a Thief, with all the game-mechanical benefits and drawbacks that come with being a Thief.
And sure, a player can disguise a character's class from the other players/characters - I've both seen this done and done it myself* - but that's all it is: a disguise.
* - best one I've done as a player was a guy I once brought in as a "Ranger"; he was in fact a multi-class Thief/Druid, but during his (fairly short) career nobody ever figured this out.
No, that's the opposite of what I'm saying.
What I'm saying is, let's say there's a class called "Priest" and a class called "Templar." A player wants to play a St. Cuthbert-worshipping character, someone that is much more interested in beatdowns than beatitudes.
But they
absolutely demand that they play a Priest. "He's a priest! That's what he does, he preaches the word of the Cudgel! Mostly by beating people in the face with it, but still. He can give as good as he gets in a fight."
"Well, if that's the case, Templar is designed to do what you want to do. They're both holy combatants, but Priests are fragile, wear cloth, and proselytize. Templars wear armor and hit stuff with weapons, so you'll get a better experience playing Templar."
"But I don't
want to play a Templar. I want to play a
Priest. Just one that fights, and doesn't do sermons."
"Okay but...you'll get all of that if you play a Templar. I guess if you build toward it you could do that as a Priest but it would take a long time and I don't think you would enjoy it very much."
"I just want to play a Priest that fights. Why can't I play a Priest that fights? This is stupid."
That is the cage of names. Locking yourself out of playing the things actually designed to make your vision for your character work,
solely because it doesn't have the right class name.
Note, though, that there are things which can look like it, but aren't it. For example, telling someone to re-skin a magic-using class in order to play a non-magic-using character is a pretty serious issue, because there really are both thematic and mechanical differences between the two. Likewise, if the alternate thing is only a partial match at best, then the difference is
not simply one of names, but of function. Telling someone who wants to play a Ranger with an animal companion to instead play a Rogue with Magic Initiate (Druid) doesn't work, because it's the animal companion they're after, not the name "Ranger," and neither Rogue nor MI (Dru) provide that thing.
So, to give both positive and negative examples (that is, ones that
are the cage of names, and ones that
aren't it), here's a few more:
CoN: "I want to play a moderately-armored, ranged and dex-melee martial character--but it
must be Fighter, don't tell me to play Ranger." (An actual issue quite often in 4e; folks
demanded the ability to play "a Fighter" in this way...even though Ranger was literally designed to do all the things they claimed they wanted to do.)
Not CoN: "I want to play a Warlord. That means someone who has a bit of healing, good ally support, etc.
Do not tell me to play a Bard and just reskin my spells." (Issue is both thematic and mechanical. Spells have many mechanics that should not apply to a Warlord's actions.)
CoN: "I want to play a Wizard whose magic is actually magic items. I don't care that there's a class called 'Artificer'! My concept is Wizard!" (There is another class specifically designed to be the thing in question, but the player sees the character only as "a Wizard," not in terms of what the character is actually
doing.)
Not CoN: "I want to be a Ranger that doesn't cast spells, but relies on his animal buddy." (An impossibility in 5e, despite UA efforts--all of which have notably failed.)
The cage of names is a serious problem in the RPG community. People are far too attached to the nametag on a class (or other things, like feats--that was a HUGE tempest in a teapot back when 4e was being worked on). They foolishly deny themselves the ability to play stuff that actually gives them
precisely what they want, just with a different label at the top of the sheet. As a result, we are forced into making several classes--like Cleric, Fighter, and Wizard--these bland, milquetoast non-entities so that they can be everything to everyone (while in truth usually ending up being not much of anything to anyone!) If we could break the cage of names, if we could collectively look at what it is classes
do, rather than merely the nametag designers gave them, we would be far more likely to have games we enjoy, classes actually designed to achieve things people want to achieve, and engaging gameplay that fits the needs and interests of most players.
Because, let's face it, as much as we love to play up how every human is an individual special snowflake, there are LOTS of pretty basic patterns and trends that can be matched. You can't get 100% of everything, but you can quite easily get 90% with a bit of effort, and more with a concerted effort, because us humans are often basic as hell.
I can't see why this couldn't be done. Sure it'd be an oddball Wizard who maybe had very few spells in its book, and it would be reliant on the DM/game to provide a steady stream of items for it to use (and then hope the rest of the party don't want those items too!); but I could certainly see a way to building something like this on the Wizard chassis.
Never minding that that's the FIRST FIVE CHAPTERS of this adventure. No. They have it in their head because that's the way someone presented it to them. People bitching about Dragon Heist not being "Oceans 11" enough when even the most facile reading of the actual module shows exactly why the comparison was being made. 
