In the shaman’s world, spirit wasn’t a thing you could possess for yourself. It was something shared like the air we breathe, and the life force we possess is shared with us by everything else in our world. Sort of like a fire, it could grow in strength and brightness or it could be depleted and sickly, but it was never extinguished. Reincarnation in the shaman’s view would just be the sharing of your spirit with your descendants and the world itself. The animals give life and they receive it again later, as does the earth itself.
Which is a circle? Yes, and everything comes full circle. This is why they preserved tribal names as well, in recognition of the continued existence of that spirit.
The reason they gave themselves what we might call animal names (and this was done in Europe also, the practice just fell out of favor there) is that they saw a kinship between humans and the other forms of life in our world. You didn’t look like a horse, but if you acted like a horse you might very well be called horse, or bear, or coyote, you name it.
Is that what happens when owners and pets start looking alike? A sharing of spiritual energy, yes.
To calm a grey hound would not kill its spirit to race? Well, in shamanism, animals were seen to have as much range of character as humans. They didn’t really subscribe to the stereotypes we do today. A coyote might be crazy, but it also might be brave. These coyotes were seen as different people in that respect.
So many creatures get an unjust bad reputation, like snakes and other reptiles, for being always evil. The Navajo saw the snake as a wise grandfather, and they believed the snake showed them how to live on the land they called home.
The search for a totem was not to search for an object or even a symbol one could possess. In their model of the world, the search for a totem was the quest to understand where one stood on the medicine wheel.
Your thoughts are welcome. Be well friends.