The "cage of names" effect is something I have seen happen as well, but is definitely something that can be worked around-- although some workarounds are easier and more feasible than others.
Playing with a home game with friends rather than an online game with strangers makes it easier to move your PC away from the generic idea of the names of your character's class or mechanics, because you know what kinds of communication is necessary to convince them your PC is an 'avenger', rather than a 'paladin' (for example). When you play with strangers and have no idea what their feelings and thoughts are on the matter-- when some of those players may be unable to not just put "Rogue" and "thief" together-- your battle is much more uphill.
I have also found that as silly as it might seem... re-writing the class progression in Word and just replacing all the terms you don't want to use with a name you do makes it easier to get others to go along with it. The Returned player in my Theros game uses the Hexblade Warlock subclass, but because their patron is not a magic weapon from the Shadowfell (like the Hexblade defaults to) but rather is the goddess Klothys... I re-wrote the advancement chart and called it a Thanatarch-- a death lord. And by doing so and always identifying the character as a Thanatarch, none of the other players have ever called him a Hexblade or wondered about a magic item patron.
This is especially true when dealing with multiclass characters-- if the player is going to MC, I usually ask them how they intend to progress in the classes before the game even starts and then write up their progression in a new chart... usually giving this Frankenclass a new name. One that is evocative of what this class combination is meant to represent. And again, players around the table who know how I work and what I put together are usually more easily inclined to follow along... whereas trying to get random players online to use the terminology would probably be like pulling teeth (and which is why I don't play online with strangers.)
And then of course at the end of the day... if we all just focus on our character's stories and personalities and desires and what they want (rather than focus on merely their mechanics)... how these characters are portrayed makes it easier to get others out of the mindset of "That's the Barbarian", "that's the Monk", and instead think of them as "That's the Brelish warforged with the blown circuits" and "that's the Firespeaker of Purphoros".
