Cats are purchased from the cat crate next to the cat trader. The crate is named greycat. If the player purchases a cat it will roam around your home base and, much like a real cat, provide absolutely nothing.
As of the patch 0.13.2f5 [December 19, 2019] Cats are an in game cosmetic that have no effect on the game; they are not upgrades.
The basic greycat costs 5 cornucopias and can be "upgraded" at the cat trader into cats of other colors at the expense of additional cornucopias, but much like the basic greycat these cats of a "higher tier" are also purely cosmetic.
According to a preview of the full release of the game, the cat trader has to be unlocked first. Way of unlocking tba.