The player character can select perks that increase weapon damage or add other effects.
Do your followers have any of these?
No, companions themselves do not have perks that you can select but they will grant additional perks if you make them happy enough. These perks are as follows:
25% faster AP regeneration if your HP is below 25%
+10 Damage Resistance against robot's energy attacks
Heal 100 HP when your HP drops below 10%
20% more sneak attack and 20% increase in Stealth Boy duration
25% more Critical Hit if you have more than 250 RADs
20% greater chance to hit in VATS
1 more guess when hacking, additionally, the wait time if you fail a hack is decreased by 50% if Nick is your current companion at the time
20% more damage against ghouls, super mutants and synths
Double XP for completing speech challenges and discovering new locations on the map
20% Damage Resistance and 20% more damage when outnumbered
20% more melee damage if your HP is below 25%
Increases Energy Resistance by 20
Not talking about actual perks in the game but as per the comments there are some unofficial perks relating to Dogmeat, in that he doesn't count as a companion in terms of Lone Wanderer, doesn't talk to you, and doesn't judge your actions. For all intents and purpose, having Dogmeat with you is like not having a companion with you while having a companion with you.
Actually yes, they have perks for themselves, mostly sneak 1-4, so they dont blow your cover. You can play around with mods to give them more perks.
I've choked on many running quests with strong. He's hilarious if you choose aggressive/rude dialogue options with people, he will chime in and back you up. His high carry weight is useful too and if you get him his heavy armor and a supersledge, he does great at keeping rushers and hounds off while you snipe. I just wish I could make him take drugs, I'd keep him high 24/7.