To unlock cats, you have to spend 30,000G or more at Mirabelle's shop. Everything you buy from Mirabelle counts for the 30,000 G requirement. Besides just purchasing animals, you could also buy Animal Medicine.
Cats have two color choices like other pets. They are black and white. The player cannot swap cats after choosing one they like the best.
The color of cat that you don't choose will go to Sprout Island and live with Martin. Each time you increase your friendship with Martin by 1 heart, a new cat will appear inside of his house. These cats belong to Martin and you can't take them out of his house.