I'm one of the cat people. I've been into cats all my life. I wasn't allowed one as a kid because a parent was allergic, but as soon as I got out on my own the first thing I did was look for a kitten. Several patient, intelligent cats later, I had been educated into proper feline etiquette and was presentable to strange cats as part of the feline community. Cats adopt the people, not the other way around. Cats will actually adopt almost anything.
Jessie, a cat I had for thirteen years, once adopted and raised a pregnant mouse. She used to cuddle the whole mouse family while the mouse mom fed her grandkittens. She had strong maternal instincts and the mouse was her close friend. The mouse had it sweet -- a huge defender against the other cats in the building for her and her offspring.
Puppy and kitten photos attest that when they grow up together in the same household, cats will teach dogs how to live in feline society. It does work that way. The cats are felinocentric as much as humans are anthropocentric.
You have to understand Cat Language first, like Dr. Doolittle. Gesture is a big part of it. Eye contact, posture, touching -- a lot of the little affectionate touches that you give your cat are translations of "you're family and I love you" moves cats use among themselves. The hip-bump translates to the long stroke from head to tail along the body.
It's recognition and affection, bonding and appreciation. Ear gestures are impossible for most humans to manage but we need to learn to watch their ears for whether they're warming up for a nice long cuddle session or warming up to play rough and try to disembowel your arm. And that's if they're in a good mood.
Don't pet the cat when it's not in the mood. Something everyone learns on the first day they spend with one, taught with half-inch long curved claws that flash out and dig very deep into furless skin.
They vocalize too though. They understand that our vocalizations have meaning. I listen to what they say to each other and it makes sense a lot of time. I can tell their mood and intent. I know the "let's play" chirrup from the "don't bug me" sound and the "I'm hungry" vs. "I'm annoyed" vs. "I'm lonely" meows.
I have heard cats attempt to pronounce my name without the consonants they can't handle. A pregnant cat whom I'd assisted with her first kittening demanded my attentions again when her contractions started -- and very clearly tried to say my name without the sounds her jaws couldn't manage.
Test this with your cat and dog if you live with one.
In a sweet baby talk voice tell the dog "I hate you, you ugly scumbag and I'd love to rip your tail off and beat you with it."
The dog will get the tone and wag its tail and want more attention.
The cat will give you a cynical look as if to say "You're nuts. What's with this stupid trick?" and not respond to tone.
Both cats and dogs recognize their names. I've seen cats who live with humans, house cats, smile. They will often do it while purring as if to say "I know that when you smile, that's how you purr, there's something wrong with your throat but I get the message." Feral cats don't smile. They get that deadpan look because the cat smile is part of the human-cat pidgin, not their native language.
I'm not sure if cat language is as symbolic as human or if they interpret it differently. I'm not sure they understand the concept of names. But it's no coincidence the word "cat" is hard little sounds that like the clicking noise you make to tell them dinner's on, cats recognize by instinct as a "come here, there's something good" noise.
An article I read hypothetized that clicking sounds like C or T sound like insects clicking in the grass, something cats in nature hunt and turn those keen funnel-shaped ears to pounce on. They pay a lot of attention to sounds they can't make.
Me, if I hear the soft crunching of someone at the cat food bowl in my room I feel warm all over and happy, I know my family's here. When I wake up to a great weight on my chest and some loud purring, I know I'm loved by a cat.
That makes the world a better place.