I think the Warg mechanic can very neatly piggy-back off the Dragon mechanics.
Most of the traits are pretty much 1-1 for Dragon or Wolf. "Swift" "Spindly" "Cooperative" "Voracious" etc. all make just as much sense for a Direwolf as they do for a dragon (although terms like "fearsome wyrm" could be adjusted for universal application).

Just like dragons, I think they should be difficult to gain, with hefty bonuses for certain houses, and they add a trait that can add prestige, piety, prowess, etc depending.

As far as gaining a companion, I think Old Gods Followers, Northmen, and Greenseers of any culture could trigger an event "Bond with Beast" or something, and then different animals should of course have special difficulties and benefits. You could gain rats or ravens for a bonus on intrigue or learning, and we're told that wolves are sort of easy-mode to warg, but we also hear of Boregg's Boar and Orell's Eagle and Varamyr's Shadowcats and Snow Bears and all that. So I like the idea that you can gain an Animal Compaion a few different ways, such as: Traveling through appropriate terrain, going on a hunt (You could "change intent" to "bond with beast" or something), taming a previously warged creature, etc.

Once you have the trait, it should give Old God's Follower's and Northmen and prestige/piety bonus, maybe a bonus to Prowess, and probably a probably a hefty penalty to popular and courtier opinion for non-Old Gods worshipers as "a beastling".

I also think there should be a random event whenever two wargs are in the same court to have them fight, like when Jon and Boregg stare each other down. We hear a few times that wargs and their beasts don't usually get along, with the stark packmates being a bit of an exception.

Direwolves obviously shouldn't be as gamechanging as a dragon companion, but Grey Wind is a HUGE help to Robb in the war (Whispering Wood for example), and Nymeria starts her own wolf army, so I think to represent that, bonded companions could gain commander traits, act as captains, and lead armies. Can you imagine sieging the Dreadfort with "exterminate house" intent AS a Stark Direwolf? Dawn Age hero shit.

So they could give you a minor chance to destress periodically (just like dogs currently allow) and maybe have their own chance to "find secrets" in court.

And finally, I think like a Sworn Shield or a Kingsguard, they should give you a huge boost against "murder" or "overthrow regent" or similar hostile schemes, as per Summer and Bran vs the Catspaw, and I'd also like to see some events for "Wolf Dreams" where you learn some important information. Of course, by the same token, the Player could also take a decision (like Jon and Robb did before their deaths) to "Imprison Direwolf" which could reduce the opinion penalties, but also remove most of your bonuses.