Basically it would be an illegal fighting pit, similar to how the Arena works. However instead of players directly taking part, you would enter your companion/pet and it would be faced off against other challengers (of equivalent power).

It would make sense to have things faced off in various groupings. So dogs, wolves, hyenas, jackals, etc would have their own group. Meanwhile dragons would face off against each other in another. Having your pet rat fight a dragon would be a little unfair. :p

Players would be encouraged to keep companions alive long enough for them to toughen up so they could try and beat out all rivals. There would naturally be gambling involved, placing bets on the winner, maybe even a more involved system, allowing bets on how long a fight might last or how much health the winner will come out with. The player could also be rewarded with various items (scrolls of familiar summoning, wands of calm monster, etc).