Is there a feat that can improve a character's level for the purpose of Rage? For example, something similar to Boon Companion but for the rage class feature?
In short, no.
While there are no feats that specifically increase the bonuses of raging as if you were higher level in official materials, there are a couple that augment barbarian rage in different ways:
Raging Vitality (Advanced Player's Guide)
While raging, you are full of vigor and health.
Prerequisites: Con 15, rage class feature.
Benefit: Whenever you are raging, the morale bonus to your Constitution increases by +2. Your rage does not end if you become unconscious. While unconscious you must still expend rounds of rage per day each round.
Raging Brutality (Ultimate Combat)
You expend some of your rage to strike your opponents with a more powerful weapon blow.
Prerequisites: Str 13, rage class feature, Power Attack, base attack bonus +12.
Benefit: While raging and using Power Attack, you can spend 3 additional rounds of your rage as a swift action to add your Constitution bonus on damage rolls for melee attacks or thrown weapon attacks you make on your turn. If you are using the weapon two-handed, instead add 1-1/2 times your Constitution bonus. This bonus damage is not multiplied on a critical hit.