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 2
Collect 1000 fishbones
Collect 200 fishbones in Level 3
Play the piano game with less than 3 mistakes
Collect 200 fishbones in Level 4
Free all the caged cats
Collect 200 fishbones in Level 5
Free Snicker
Find Nala's cup
Find Nora's score
Free Abb
Free Sheldon
Find Cooper's book
Find Oskar's ties
Jump over chihuahuas 10 times
Throw 20 apples
Collect 200 fishbones in Level 1
Hit dogs 10 times
Break 10 vases
Free Tonks
Throw 5 rats into the graveyard holes
Free Daisy
Find Henri's book