IMO what the identity of the Fighter has traditionally (i.e. - not just 5E but through the history of D&D) been is sheer training/skill + mastery of all armor and weapons.
For armors, they almost there with Fighters and Paladins being the only ones with full proficiencies, although there are other ways (races, feats, subclasses) to get the missing profs.
For weapons, they could have tried giving proficiency in all weapons only to Fighters, and to give Paladins/Rangers/Barbarians only simple weapons + 3 martial weapons of choices. But I am sure this would not have been a popular design decision.
Another option would have been to grant Fighters (only) proficiency in also all weapons beyond simple/martial, including both improvised and exotic weapons. Alas, there are no exotic weapons in 5e at all at this point.
That's about proficiency rather than 'mastery', but how to represent the latter? I see two options:
Mastery as efficiency > Attack or damage bonuses. IMO they purposefully stayed away from additional attack bonuses as much as possible, to stay more faithful to the idea of bounded accuracy. That's why we don't have any 'weapon specialization' around in 5e. OTOH, damage bonuses (as in 3e Fighter-only Weapon Specialization feats) could have easily been there... Anyway I think that "Fighting Styles" are the closest thing to this, except that they don't focus on
one weapon but on
one weapon category/setup.
Mastery as flexibility > Being able to 'unlock' additional weapon abilities. IIUC the design decision was to implement this within feats, to make these available to everyone, and then just give more feats to the Fighter.
As usual, there is a strong trend to react to the Fighter's stuff by saying "why shouldn't my Ranger/Paladin/Barbarian/Rogue/Cleric be able to learn that too?". This is what risks killing the Fighter class in every RPG. I am happy that at least this is not completely true in 5e!