The game won't clarify you how your dragon pet works. You have a egg in your back which evolves into a Dragon Pet as long as you give him healing items that have "Growth (Dragon Only)" that is what makes it evolve and at the first steps the dragon will aid in you battle.

Blaze is the name of your pet. Blaze, evolves at 10HP, 50HP, 150HP, 300HP, and 600HP. His HP cannot increase beyond 600!, so DO NOT feed him anything after that unless he is injured.

Perks: At 10 HP, he can only use a small fire breath. At 50 HP, he can now use physical attacks, and has a bigger fire breath. At 150 HP, his physical attacks are more powerful, and he now has triple fire breath. At 300 HP, higher physical attacks, and his fire breath affects a wide arc. At 600 HP, higher physical attacks, and his fire breath will attack a massive arc that hits almost everything.