"Very own" can have a couple of different nuances, and it always adds more emphasis on the ownership that just "own".
It tends to be used after someone has recently bought or acquired something expensive or something not typically owned by the average person. Or sometimes when a young person acquires something for the first time, such as their first car.
"The millionaire has his very own beach" - i.e. it is a private beach belonging to him
"He has his very own ice skates" - i.e. they belong to him, whereas most people just hire them.
"She's just got her very own flat" - i.e. she's living by herself having moved out from her parents for the first time.