Here's a brief guide on how to equip Dogmeat, Codsworth and your companions with stylish items.
Do take note this is only a choice of style since, afaik, even his dog armours have no protection values. This changes if we talk about companions/ settlers since giving them armour or weapons makes them much better.
First things first, point to Dogmeat (or anyone else you want to dress up/ equip) and hit E to open the command menu, then go to Trade.
Place any apparel item you want on his inventory, once they're there hover on the item you want him to wear and hit T to equip/ unequip.
There, you're done!
As said in the title you can do the same with other companions or even settlers if you want to give them new clothes, armour or weapons.
What and Where
Collars: seen as a normal dog collar or a chain one, usually on raider attack dogs
Bandanas: seen in green,gunner's camo,red or skull patterned so far, found somewhat rarely on raider or their dogs, also on gunners in the city
Dog Armor: both in normal or light variants, I only found one of each so far while clearing a fortified raider encampment after killing two of their attack dogs. Provides no damage resistance sadly
Welding Goggles: found on raiders or just in workshops and similar shops
Muzzles: as usual, on raider attack dogs
Hats: The only one I've made successfully him equip so far is a Bowler Hat
FOR ANYONE ELSE (COMPANIONS/ SETTLERS)
Feel free to give them any armour/ weapons (don't forget to give them ammo aswell) or hats you wish: You can spend time customizing each of your settlers.
Want them to look like rugged farmers or a militia? Now you can. Want them to go around in a red dress and sunglasses? Just look at the screenshot below, now you can too.