To make the option to Select Existing Address available, review the shared address to determine who is currently sharing it. Then ensure there is a relationship between each of those constituents and the constituent you are adding the address for.
To determine which constituents are sharing an address, hover over the sharing icon Share Iconnext to the address type. 

Note: The same conditions apply when adding a phone contact. In order to add an existing shared phone number, the constituent must have a relationship with all other constituents already sharing it.