Domestic cats are different colours because of their domestication. Although, to be fair, among the wild cats there are subtle variations in coat colour such as occurs with the bobcat for instance and the African golden cat. But these as mentioned are subtle and natural evolutions.
The thing is this: there was a time about 10,000 years ago (believed) that there was only one domestic cat coat colour: tabby brown/grey. That was it. If you wanted a domestic cat there was only one model on offer. It was like the buying Model T Ford: black and sit up and beg.
The reason? The domestic cat was a freshly minted African-Asian wildcat and all of these wild cats are the same: brown/grey tabby cats.
So what happened next to create all these colours? People intervened. There would have been some slight changes in the coat from a striped tabby to a blotched tabby as a result of natural selection and genetic mutations.
Humans would take a fancy to strange new appearance and informally mate one cat with another. This was thousands of years ago. They were not neutering and spaying cats in those days. There wasn’t a single veterinarian.
Gradually over the first 9,500 years of cat domestication through informal cat breeding some coat colours emerged such as the bicolour and solid colours in stark contrast to the natural camouflage of the original tabby.
Then in the middle of the 19th century the cat fancy emerged. It started in England and the cat fanciers fancied creating fancy cat breeds! The first were the British Shorthairs, Persians, Siamese, Abyssinians and Turkish Angoras. This was formal cat breeding.
For the English the British Shorthair was there already in moggy form, imported into the country about 1,600 years earlier by the Romans. All they had to do was to refine the appearance; polish up the raw material, the humble moggy, and make him and her look smarter. They achieved this through selective breeding. This is controlled inbreeding of cats they liked the look of to make more of them and in a refined appearance.
To return to the question in the title. Domestic cats have different colours because people intervened in natural selection and through artificial selection created new coat colours and patterns. It is all about using genetic mutations to good effect.
It is about picking up on a genetic mutation and enhancing and strengthening it to produce a more distinct appearance. Yes, it is like creating a new look car with bright colours and a modern shape.
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