In this guide you can find some useful tips on how tame creatures.
Guide to Adopt or Tame Creatures
You need to repeat the pet/feed/scratch moves until the ring is full (yes that means coming back on different days) once the ring is full you can (left click) to adopt the creature and have them live in your nature reserves.
If the ring is full and you don’t get a prompt to take a selfie or adopt, try changing your active tool to the gloves. If it’s the net or something, the prompt doesn’t always appear.
Note: Pet house is a addon for your small and large houses when customising from what ive come across its not the most common item to find while gardening.