First post here, so "hello"
Actually i think a good compromise between personality e customization left to the player will be the following:
Companion will earn some gold from the loot, which they can use to buy some basic improvements when i'm visiting a shop just to avoid having some companions with crappy equip. This will be rare while you level up i can suppose due to the better loot you find. Of course that items can't be taken away from the Companion but just replaced with the equip you choose to give to them. And should be equipped back if you take back the items you've given to them.
Companion should complain if you make certain decision about who receive an item. They shouldn't decrease their relationship with you if you just equip a good items on someone else. You're the boss anyway, but maybe grumping a bit if you give a good bow to the thief instead of the ranger. This will help feel like the companion are not just some bunch of pixel. This will help newbie too, so they can understand when they're messing up with the equip optimization.
Companion should have at least a peculiar trait. A drunken dwarf, a self-conscious half-elf and so on. This traits will affect the gameplay. As someone suggest, a drunken character without alcohol will be rude and distracted maybe, gaining some malus to both social and physical skill.
Instead, about the feature of "Companion can gift something to the player". I think this sould happen only when the relationionship is good enough. And should be something related to the Companion. I don't think this features will be easy to implement not for technical issue but for the logic behind.
"Why the hell <Companion> should gift me a powerful items just now, after we've fight side to side against any sort of monstruosity risking our lives? Can't him just use that items itself or give it to me before?? Jeez such an idiot!" - I suppose this explain why i think this is a nonsense.
Of course this is only some point i find interesting and must be integrated with some good interaction with the companion meanwhile the story goes on (See Dragon Age: Origin for a good example imho).
That's just my 2 cent.
Have a nice day.
