You may want to post this to the Tableau Ideas section. Some options might be allowing a user to define their sort favorites by:
- sort A-Z (asc/desc)
- sort manually
- sort by most used
- sort by most recent
Aside from this functionality being built-in to Tableau Server, you could query the Tableau postgreSQL with a user filter to display a users' favorites. You could perform additional functionality for the sort (except for manual) as mentioned above with additional querying.
Not sure if the REST API will allow you to query and expose users' favorites, but worth exploring.