Does wearing full armor set grant bonus?

The only armors that have a sort of "set bonus" are in the DLCs. All other armors only have the perk bonuses (+25%) for a matching set.

From Dawnguard:

• Dawnguard Light - 25% less damage from Vampires and Drain Life spells
• Dawnguard Heavy - same bonus

From Dragonborn:

• Ahzidal's Armor - 10% Enchanting with the boots + exactly 3 other items from the set of 7. More or less than 3 other items will not activate the bonus.
• Deathbrand Armor - each piece's enchantment gets stronger when wearing more pieces of the set

Also from Dragonborn, but weapons:

• Soulrender and Bloodscythe - absorb health and magicka, chance to reduce armor and dispel magic when used together

No, there are not armor set bonuses in Skyrim.

The closest thing is a range of perks that will improve various attributes when wearing a full set from a particular category. This wiki page details this towards the bottom of the page.

However, perks obviously don't affect your companion.

Some followers have perks. These are not listed on the Elder Scrolls Pages, generally, but a GameFAQs forum user was kind enough to use the Creation Kit to identify and list them. This list is four years old, so it could be out of date if patches have changed things.

Based on that list, there are eight followers that have perks that give them bonuses for full sets of armor:

Well Fitted

25% armor bonus if wearing all Heavy Armor: head, chest, hands, feet

Golldir

Tower of Strength

50% less stagger when wearing only Heavy Armor

Golldir, Mjoll

Custom Fit

25% armor bonus if wearing all Light Armor: head, chest, hands, feet

Belrand, Karliah, Mercer Frey, Celann, Durak, Serana