I’m not sure what you mean by “really” living. Is there fake living? Does one need to be aware of mortality in order to really live? What kind of awareness? Must one think about it all the time? Or once in a while?
I don’t think living has much to do with your awareness of mortality. It has to do with how much attention you pay to what you are doing; to what is in front of you. In other words, it is about being present and mindful. Thinking about death is the opposite of mindfulness, so I’d say that someone who is constantly thinking about death is more likely to fake live than really live.
Ownership is another one of those bogus concepts. What is ownership? It’s a kind of societal agreement that we can mark territory that no one else will violate without our permission. Territory can be land or things. It’s a social agreement. It only works as long as everyone agrees to follow that agreement.
But ownership is always temporary, and lasts as long as we can maintain control over something. If you are afraid of losing it… I guess I don’t see what that has to do with real ownership compared to fake ownership. You could say we own things as long as we use them. If we stop using them, eventually someone else will want to. You may not care to stop them. Hell, you might give it away.
So, no. You don’t need to be afraid of losing things in order to own them. The concept of ownership is not clear, anyway.