How do cats see humans? Do they think we are just a big cat?
The strict answer is that we don't know. We can't look into the mind of a cat, let alone a sampling of a statistically significant number of cats.
However, the big take-away I've always gotten from my interactions with cats is that they are very different. Some are just very affectionate by nature, and come right to people expecting to get a scratching behind the ears. Others are less demonstrative, while others may be very shy of all humans, even their chief feeding human. I get the impression that it is fairly common for a cat to identify its chief feeder, and allow that person to come to them and pet them, and they may also go to the person, while avoiding all others.
It appears that cats have this social connection with their mother when a kitten, and it is common that they transfer this association to the human who generally makes them feel cared for and safe. Generally this would be the chief feeder, but not always.
I happen to know a couple fairly well. They are both gentle and loving people, and they both strive to treat their cats with gentleness and respect. Nevertheless, it is my suspicion that the man is just a bit more careful, hesitant and gentle with his contact with the cats, although the woman certainly is not abrupt or coarse with them at all. They have four cats, and one of these came into the house as a kitten, and remains a very small cat, and is very skittish. She has definitely chosen the man without any doubt. She comes to him and asks for a snuggle all the time, and is very affectionate with him. I suspect that she has picked up on this gentleness of the man, although of course one never really knows what is going on in an animal's mind.
The simplest explanation of the choice of primary human is probably based on stuff we barely know anything about, perhaps related to scent as well and sound and touch. Some mysterious quintessence causes one cat to choose one, another cat to choose another. And of course, since cats are creatures of habit, once the choice is made, it is unlikely to change.
I've read of some reports that have cats leaving kills for their human, even in one case leaving a kill on the pillow next to a highly depressed man, causing him to (1) have a major "Yechh" response, then (2) an epiphany that the cat thinks he needs to eat, which caused the man to get a major laugh out of the event, which lifted the worst of his depression. Or did the cat simply care about the depression of his companion and just do the only thing he could to try to make the person feel loved? I guess we'll never know.
I remember one time when I had a severe migraine, all of the cats in my entire house (4 of them) very gently and quietly got onto my lap and just lay there motionless, just purring gently. I had been drifting in and out of sleep, and never was aware when any of them got onto my lap and chest. I simply became a bit more alert, and there they were! I felt the good wishes of them flow over me. It was very comforting. It was intriguing that while these cats got along fine, they generally did not hang out with all of the other cats; they had a favorite, and they didn't seek out the others, so the four of them laying right up against each other was a major step for them. I think they set aside their boundary issues because I was obviously sick and they wanted to try to help. It was the only thing they could do, and I'll never forget it. It also convinces me that cats have genuine feelings for others, and can demonstrate it.
On the other hand, next to any cat, even the largest of the large, a typical human is many times their size, and so they are understandably cautious. One of my cats knows me and will get in my lap and settle down to snooze, but if he's eating (and he loves to eat), if I move through the room, no matter how slowly and softly I tread, he still runs away from his dish. It's gotten so I try to avoid going into the room after I have put down the food, so as not to startle him. After all, if a creature as tall as an six-storey building suddenly started to move toward me at all, I'm sure I'd be scared sh*tless.