There are a couple of things you could be referring to when you ask about the Japanese and cats. One is the Maneki-neko, the Lucky Cat, and the other is the traditional guardians of Buddhist temples in any country.
To begin with the Maneki-neko is sitting at the entrance to many businesses with his paw up. The position of the paw is a welcome to customers meaning to come on in. Some of them have the other paw up, and that is a good luck charm for them to get more money. Some Japanese keep these at home too for good luck too. They are kind of like the rabbit’s foot in the west. The picture I found is the original colors of the Maneki-neko. However, over time, they have begun making them in all different colors, but they are still Maneki-neko even if they are other colors.
There are several stories as to how these cats came about. Some say he is related to the Chinese lucky cat. Others say that he comes from Japanese Buddhism. There is a story about a Buddhist priest who was standing under a tree during a bad rain storm. He saw a cat across the way who seemed to have his hand up beckoning to him. It surprised the priest because cats don’t usually beckon to human beings. Even though it was raining really hard, he went out into the rain to check the cat out. When he got away from the tree, lightening struck the tree and the tree came down saving the life of the Buddhist priest. After that, the Buddhist priest was very grateful to that cat and made him very special. He was supposed to have been the first Maneki-neko.
Another cat you could refer to is the guardians of the Buddhist temples everywhere. In almost any country, including Japan, you can see lions guarding the Buddhist temples except in Korea where the Haechi, a mythical beast similar to a lion, guards the Buddhist temples. In Thailand, huge cats used to live in the Buddhist temples as guardians. They say the Siamese cats are the descendants of those cats from old Siam, Thailand, and that they just slowly, through time, became smaller. All Buddhist temples are guarded by these statues. I am left to understand that at one time, live big cats actually guarded Buddhist temples.
It seems to me from what I understand, the love of cats probably came from Buddhism in Japan. Perhaps the Koreans put Haechi at the entrance to their temples because many Koreans for some reason, don’t like cats. The Korean dislike for cats could be because their country was overrun by tigers, and the tigers were man killers. The Koreans and the Japanese hunted the Korean tigers to extinction. There was a time that a sign of being rich in Korea was to have tiger skins. Now a days, the Koreans like to romanticize about how beautiful the tigers were, and the old Korean folk tales are full of stories about tigers. As for the Japanese, to them, the cat is a bringer of good fortune.