Feeding your pet fish, which can be caught in fishing holes in towns or dungeons or purchased from vendors, will transform your cat or dog into one of the creatures you encounter through the game. This has an effect, either positive or negative, on your pet's base stats.

Here is what each fish does:

In order to get the Dream Pet achievement, you have to use each of the fish (excluding Dogfish) to transform your pet at least once through the game. The duration of the effect does not matter.

The final transformations come by giving your pet the potions granted by restoring the statues in the Temple of Fate. Gathering the items needed for the hero statues in each of the dungeons and restoring the items to each statue grants the player a potion, which then transforms their pet into the form of the hero in the statue.

The required items are:

Dungeon of Typhon

• Argonaut's Ring
  
• Argonaut's Buckle
  
• Argonaut's Helm
  
• Argonaut's Sword
  
• Argonaut's Shield
  
• Argonaut's Chain

Dungeon of Druantia

• Swathe's Ring
  
• Swathe's Buckle
  
• Swathe's Helm
  
• Swathe's Sword
  
• Swathe's Shield
  
• Swathe's Chain

The items should be somewhere in their respective dungeons, within the first 28 levels. Whenever you enter a level where one of the heroic items are hidden, you will be aware of the item's presence:




And in your inventory, the items are marked with a capital H, so hopefully you don't accidentally get rid of them:



Note: I actually got the Dream Pet achievement after restoring only one of the statues and giving my pet the potion, after having used each of the fish at least once as well.

You should also earn the Prince of Thieves achievement once you restore both statues, since it requires that you "Recover and return to the statues all the stolen equipment of the legendary heroes from Druantia and Typhoon."