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