So I was skimming through the Barbarian Class and saw that some feats, namely Weapon Specialization, mentioned an extra +2 damage if you had legendary proficiency in a weapon. However there seems to be no way of achieving that. Barbarian weapon proficiency caps at master. Am I missing something?
Nope.
They get other offensive bonuses instead, mainly raw Rage damage.
You are not missing anything. The text appears to be there in case future options make Legendary Proficiency possible.
As others said, you are not missing anything.
You may find Gisher's Guide to Proficiency Bonuses useful.
That said, in the Playtest there was an archetype (Grey Maiden) that allowed you to gain Legendary proficiency with shields and Grey Maiden plate. I would not rule out the possibility of there someday being an archetype that allowed better weapon proficiency with a very limited set of weapons.