The Actual Owner is a sidequest in Star Ocean: The Divine Force you can pick up at the Seaport Of Rythal.
If you head to the medicine center house at the Seaport Of Rythal, you can speak to Doctor Vahnel to begin the quest.
He will ask you to deliver a cat called Sumi to it's own in the Imperial Capital.
Head to the Imperial Capital and speak to Elderly Man With A Cat and deliver the cat.
Then return back to Doctor Vahnel for your rewards.
Reward: