When I access a shared Contacts folder from another user, I don't see the "Outlook Address Book" tab in the Properties dialog of that folder.
How can I add a shared Contacts folder as an Address List to my own Address Book?
Sadly, adding a Contacts folder from another mailbox to your Outlook Address Book is not supported. You can search for the address directly in the shared Contacts folder or consider putting these contacts in a Public Folder or SharePoint List you both can access.
To be able to add a Contacts folder from Public Folders to your Outlook Address Book, you’ll need to add it to your Public Folder Favorites.
To address a message from within a Contacts folder, select the contact(s) you want to send a message to, right click on your selection and choose; Create-> E-mail
A shared Contacts folder doesn't hold the Outlook Address Book tab.
Sync add-ins for shared folders
Another way to go would be to sync the shared folder with an additional Contacts folder in your own mailbox via an add-in such as CodeTwo FoldersSync (free) or a more advanced alternative such as CopyFolder.
As the Contact folder will then exist in you own mailbox, you can set it to show as an Address List in your Address Book.
CodeTwo FoldersSync is easy to set up; create a new Contacts folder in your own mailbox and open the add-in’s settings dialog to set up a folder pair. To synchronize, click the newly added Synchronize button on the Ribbon (Outlook 2010 and Outlook 2013) or Toolbar (Outlook 2007 and previous.
CopyFolder is similar to set up but also allows you to filter the synchronization process based on categories you select and automate the sync process.
Outlook 2010, Outlook 2013 and multi-Exchange account support
If you have Full Access permissions for the mailbox and have not also been assigned as a delegate, then you can also add the mailbox as an additional Exchange account.
If you are also a delegate for this shared mailbox, then this method could lead to issues with processing meeting requests and cancellations and is therefor unsupported.