Edit: There is now an entry for this in the ArcGIS Ideas area. If you want it, vote it up... Search, find, and use ArcGIS Online items in Portal for ArcGIS.
How can our organization's Portal for ArcGIS users search, find, and use publicly shared ArcGIS Online items within their Portal for ArcGIS environment?
ArcGIS Online Content
Portal for ArcGIS also provides access to all nine basemaps available from ArcGIS Online: Imagery, Imagery with Labels, Streets, Topographic, Terrain with Labels, Light Gray Canvas, National Geographic, Oceans, and OpenStreetMap. Optionally, premium content from ArcGIS Online can also be used with Portal.
So there's that - basemaps and premium content.
Short of adding, describing, and styling each publicly shared ArcGIS Online item to Portal, have you found a way to configure Portal to return publicly shared ArcGIS Online items when a user searches for them?
We're at Portal 10.3.1.