To answer the second question, I define valuable as worth. So if we describe something as valuable, it means it has great importance to us.
As for the first question, it depends on what our possessions are. For example, Person A could have a mansion and several sports cars and say that their money is their most valuable possession. Here, it can be interpreted that Person A is materialistic and vain.
Another example would be how Person B could come from a big and loving family and say that their most valuable possessions are the memories of a blissful childhood. In this case, it could be interpreted that Person B is caring and has strong family values.