One might think that the biggest dog or cat would be the "alpha" (I understand that this term as applied to wolves is controversial and observers did not at the time understand that the alphas were simply the parents of the other dogs and so naturally were dominant -- there had been no fighting or whatever to establish this.) in a household but people who have had multiple dogs have told me that a small and much older dog was the "boss" whatever that meant, maybe it ate first or something.
I have seen with cats that size makes no difference (maybe): a larger cat very much was submissive, would be chased away by the smaller cat who frankly was the favorite of the humans in the house. However, somehow between the two of them they had worked out a process for eating where indeed the larger cat ate first with the smaller (and BTW younger; she had come to the house months after the first cat, a dumb idea -- I think adopt two cats at once, maybe siblings -- the first cat very much understood the implications of the new cat, they just never liked each other although occasionally for warmth maybe would sleep together) cat waiting a meter or so away, in the same spot as far as I could see each time.
I don't know if this was a compromise where the small cat had gotten the rights to sit with humans and the large cat, who liked eating, got the right to eat first. I don't believe they ever actually discussed this (but who knows?) but they seemed to have an agreement that they both observed -- the larger cat would still try to sit on human laps but could easily be chased away, she would not fight back even after being bitten in a subtle maneuver that began with grooming.
Funny creatures, cats.
But it is hard for me to believe that at least sometimes the hierarchy is not settled via fighting in which case size would probably matter -- perhaps more so with, say, a group of "outside" dogs.