As far as I know, there’s no approriate equivalent for CardDAV (address books). Other possible mechanisms would be a group principal membership, but that’s currently very wide-spread.
Short answer: It’s a CardDAV limitation.
I have a Fastmail family account. This comes with a shared address book that all accounts in the family have access too; however, when I sync with DavDroid, only my personal address books are shown.
I contacted Fastmail and they replied with the following:
“Some clients don’t detect the shared address book correctly. So you would need to manually configure them.”
The reply also included instructions for manually adding the shared address book. This included a server address and username specific for the shared address book (which I don’t want to post here). There is an option to create a new address book with DavDroid, but I can’t add an existing one manually.
DAVDroid version: 1.0.7 (98)
@rfc2822 wrt 1) The FastMail shared address book is returned with the other address books that are available for the specified user using standard CardDAV/WebDAV interactions e.g. addressbooks collection via resourcetype, addressbook-home-set or current-user-principal. What methods does DAVdroid use?
@rl All of them. DAVdroid tries to detect the current-user-principal, then the addressbook-home-sets, then detects whether either the home sets themselves or their member collections are collections of type addressbook.