The Cat Returns/WMG

It is possible to stumble upon the Cat Kingdom by accident
The reason there is humanlike culture there is that people have done this, and been permanently transformed into cats before finding or hearing about the exit. These humans then worked to construct human infrastructure and/or explain how it works.

Haru's a mental patient.
The whole thing is just her hallucinating. The Cat Kingdom is how she sees the asylum, her transformation is how she views the therapy process, the Baron is an imaginary construct created by her, and her eventual escape is self-explanatory.

Alternatively, Haru WAS hit by the bus.
And the rest of the movie is her Dying Dream as she lies in a coma in the hospital. The cat's world is actually the afterlife, and the more she stays there, the more her real body continues to die. In the end, her escape from the kingdom coincides with waking up from the coma and returning to the real world.

All cats in the human world are outcasts/deserters from the Cat Kingdom.
In the scene where the Cat King arrives to thank Haru, the cats living on the street are clearly not happy to see him. Muta is a criminal in the Cat Kingdom, apparently having left by choice after wreaking havoc, and he doesn't like the king. All the cats of the human world have seen through the Cat King and left to raise their families free of his influence.

Only certain cats have the ability to speak.
When we first see the cats that live in the Cat Kingdom, specifically the light pink and blue couple, we can only hear them meow instead of speak. This might be because it's a gift acquired after living in the Cat Kingdom for a while, or only those somehow connected to the Royals, be it blood relation or servant to them, are able to speak. While Muta can speak, we don't know his past other than eating all the fish, so either of these theories could be correct. Baron's a Creation, and not an actual cat, so he doesn't count.

The subplot of the story is the collapse of Shizuku and Seiji's relationship.
Let's see, young girl due to the involvement of a magical cat, is promised in marriage to a handsome prince? Sort of the plot of Whisper of the Heart, eh? Except that it turns out this relationship requires her to transform into something she isn't, and in the end she escapes from it, returning to reality. If Haru's story is written by Shizuku, it would seem to be a metaphor for a not very happy ending to the first film.

The Cat King has apparently gone insane after his Queen died.
We never saw Prince Lune's mother so she must have passed on some time ago and this devastated the King very much. His attempt to get his son married as soon as possible is probably because he wanted another woman in the family to love. Not implying anything here but he thought it would make him happy. That's why when his son rejected Haru, the king tried to marry her instead.