I do not understand the exact meaning of this statement: "Detachment is not that you should own nothing, but that nothing should own you."
This seems like a Buddhist idea. An item owns you if you invest time and resources maintaining it and even possibly define yourself by those items. You can have things but you should be ready to let go of the items, like a mandala.
The reason for this idea in Buddhism is the belief that excess desire is the root of pain. All things in life are temporary, you can appreciate them while they exist but it hurts you to cling to things that cannot be.
The first half of the quote is fairly straightforward.
Detachment does not mean that you cannot own material possessions
The second half requires a slightly more artistic interpretation, but in this context it effectively means that
the acquisition and possession of material possessions should not be your chief priority.
Googling your quote shows a related quote:
Detachment is when the dunya remains in our hand – not in our heart.
This phrase in simple terms just means "Do not get attached to anyone or anything, to the extent that you cannot survive independent of it or them".
When it is in reference to a person, it means you may love someone dearly and have them in your life, but you should not allow your life to revolve around that person, making it seem that they control your life.