Cats can be found via the Old Village exploration card, no breeding.
You can make a small mod to allow cat breeding. Adding a new recipe for the existing cards is trivial. Here are mods for cat breeding: first one for “cat + cat”, and second one for “cat + milk”:
https://drive.google.com/drive/folders/1mfc8dufvpALFjVRp7QVJ2KKm6BUIq09d?usp=drive_link
Load these using the “manage mods” button in the main menu, activate, and enjoy :)
This game has a built-in support for mods. Now, mods are JSON files that patch game core data with new cards, recipes, etc. See the files at the link above - these are plain text files that contain the recipe you requested. Click on the “manage mods” link at the main menu, then click “load mod” and upload the JSON file. Then click “activate” and “go back to menu”. That’s it, your game will now have cat breeding recipes :)