I have a daughter and numerous pets, and I consider the pets every bit as much my children as she is.
They are, in fact, pretty much eternal toddlers. They will, for all their lives, be dependent on me to provide them with food, shelter, medical care, grooming, and discipline. Unlike my daughter, they will never be self-sufficient.
My daughter considers her pets her children, and me their grandmother. In fact, she calls me Granny when telling them to go to me. And that's fine with me.
ETA: Nervy Girl, I don't know where you shop, but I see cards for step-parents in the grocery store on Mother's Day and Father's Day, as well as birthday cards specifically for step-parents.
As for people buying themselves cards from their pets, it's usually other family members who do that. One year, my daughter, in addition to the gift she gave me, bought a card for me from all the animals (mine, hers, and the fosters), and signed each of their names to it, each in a different penmanship. I thought it was a very sweet gesture on her part.
ETA:
Momof1, I have been out in pre-hurricane storming, looking for one of my cats. Turned out that one of the neighbors had already brought her in. Would I give my life for one of my pets? Yes I would.
ETA:
The "fire question." In the even of a fire, my daughter and I would BOTH be loading beasties into the car as quickly as we could to get them to safety. And I would tell her to take the pets out first even if I needed help to get myself out.