Have you ever rescued a cat or dog from a shelter? If you get lucky and find the right one, it becomes part of your life for what is hopefully a long time. You look forward to seeing it, and it looks forward to seeing you.
It's not that you expect to come home to a cat or dog. It's that you're coming home to that cat or dog (or whatever you like). You didn't adopt an animal, you adopted that animal.
I imagine it works much the same on the other side. They are happy to have been recognized as a specific someone, and not just a cat, or a dog, or a generic pet for that matter. There's no "pet spot" in your life. It's a spot for exactly them.
Now then, have you ever been on their side of that? Have you been somewhere that expects you to be there, and not some generic creature which does the things you do? It's a place where they go looking for you and only you, and are delighted when you show up.
That sounds like a really good situation.