In the official guide of the game, it states that if you feed a dog three fruit or meat items it will lead you to a hidden chest in the area.
I also found this link where it explains dog behavior and rewards.
How to Increase Bond With Dogs:
- Stand near a dog. Just face a dog and look at it — a puff of pink
will appear. That’s the first step in any doggy interaction.
- Give the dog food! It doesn’t seem to matter what type of food —
though the quality will increase the bond. To give a dog food, open
your inventory and select “Hold”, then drop the food on the ground
near the dog. If the dog sees you drop the food in front of its
face, you’ll increase your bond.
- [Unconfirmed] Run! After increasing bond, the dog will follow you
around. Do some sprinting with the dog and let it chase you.
- [Unconfirmed] Play fetch. Use any wooden branch cut from a tree and
throw it while the dog is paying attention. It will run off and
retrieve the stick right away.
I can confirm the 3rd one works, but I've never tried to play fetch with the dog.
After a coupel hours of testing and rising friendship with a dog I was not able to reproduce the fetch functionality. And confirmed that the official guide doesn't mention it so I would proced to assume it's not actually in the game.