Cats are present as a minor part since the beginning of the franchise.
The cat is only once obtainable through gambling when there is the event "The Gambler" that can trigger after Gustav comes to the fort. If you win the gambling three times in a row, he will say that he doesn't have any more supplies to even feed his cat, so he gives her to you. You can count on the fact of winning her if you play repeatedly and you got enough food in stock.
Just as with dogs no cats can be exchanged with Gustav. Animals do not belong to his assortment.
They are randomly found and taken to the base after succeeded kill-missions. They raise general happiness for +5 and differently than with dogs this bonus won't be taken away again (as happiness raised by events, not comparable with happiness raised by buildings).