Classes are just bundles of mechanics.
Yes.
There is nothing stopping the designers from designing every class bundle such that they share no mechanical commonalities with any other class at all.
As an example, we can make each skill it's own attribute. Then we can have as many attributes as there are skills we can dream up. We can ensure that no 2 classes ever share even 1 skill in common and that players of said class have no way to increase their skills in non-class skills. So what deeper mechanical commonalities do the classes in this hypothetical system share? Why are we limited here to 4 classes before we have commonality?
Heh, really?
So, only one class has Hit Points, and no other class has Hit Points (Strength-Fortitude)?
Only one class has AC, and no other class has AC (Dexterity-Reflex)?
Only one class can cast spells, and no other class can spellcast (Intelligence-Perception)?
Only one class can resist mental assaults, and no other class can (Charisma-Will)?
...
Only one class can use swords?
Only one class can use bows?
Only one class can wear armor?
Only one class can Stealth?
Only one class can know History?
Only one class knows how to Jump?
...
Whether such a roleplaying game is even theoretically possible, the D&D roleplaying game has classes that share many mechanics in common with each other.
With balance in mind, I support player customizability, and a classless class.
At the same time, I dont find "Atonal Music the RPG" a fruitful way forward for D&D traditions.