Here is the full list of all 23 The Purring Quest achievements. It takes around 3-4 hours to unlock all of the achievements on Windows.
Collect 200 fishbones in Level 3
Collect 1000 fishbones
Play the piano game with less than 3 mistakes
Find Nora's score
Collect 200 fishbones in Level 4
Collect 200 fishbones in Level 5
Free all the caged cats
Free Snicker
Find Nala's cup
Hit dogs 10 times
Break 10 vases
Free Abb
Jump over chihuahuas 10 times
Free Sheldon
Find Cooper's book
Collect 200 fishbones in Level 2
Free Daisy
Find Oskar's ties
Throw 20 apples
Collect 200 fishbones in Level 1
Throw 5 rats into the graveyard holes
Find Henri's book
Free Tonks