Yes. If I have a hammer throw paladin and I want to make a warpath paladin, I create a new character. So not only do I have more than 1 character in the same class, I have more than 1 character in the same mastery.
Other than some build defining unique, not really.
If you don’t select any skill and use only your base attack, you won’t get far even with 4LP legendaries.
But I think your misunderstanding regarding this comes from different definitions of “character identity”. Character identity isn’t really the character personality, motivations, flaws, etc.
Character identity is simply the choices you make to create that character.
In a regular RPG like D&D, this includes choices like the aforementioned personality, motivations, flaws, etc, as well your race, your class, your height, your hair color and so on, and along with it your skills, spells, etc. All of these are sort of permanent, baring jumping through hoops to change them.
In a diablo-like, you have less choices, but they’re still choices. You choose your class, your mastery, your skills and your passive tree. You don’t have choices for your personality (though you’re free to come up with them if you want), but you still have choices. And those are part of the character identity.
Making those choices not have meaning because you can undo them with a touch of a button removes that identity.
And making it so that switching a mastery is a 1-minute thing while releveling a new character will take lots of hours, depending on which level you are, also kinda removes the incentive not to use respec.