Do Cats Perceive Humans as Bigger Cats?

It is thought that since pet cats knead their claws on us, lick us, or bring us their prey, they view us as if we are a valuable companion in their lives. Does this mean they perceive us as a big and weird looking cat?

I think the answer to this lies in another question: When we kiss our cats, talk to them, and play with them as if they are a valuable companion in our lives, do we view them as small and weird looking humans?