Here is the full list of all 15 The Princess, the Stray Cat, and Matters of the Heart achievements.
You had a flashback of your house on fire.
You began the first optional route.
You watched the opening movie sequence.
You have become a cat.
You began the optional section of the heroines.
You began the Patricia route.
You completed the Patricia route.
You began the Michi route.
You completed the Michi route.
You began the Shachi route.
You completed the Shachi route.
You began the Yuuki route.
You completed the Yuuki route.
You began the Firecracker Hero Nobuchina's adventure.
You ended the Firecracker Hero Nobuchina's adventure.