Media Bookmarks

I'll bet you are storing a lot of dupes of some pictures and videos.
It would be nice if a user could 'bookmark' a media item, then have the ability to create named albums of bookmarked pictures. The original user would still hold their media item, and have some measure of control over it (it could vanish from my bookmark album). This could work for both public and privately shared media.
There's probably no stopping users who will still download and save their own copies, but some users may opt to only bookmark what they would like to revisit conveniently.

Something like a "Liked Media" section, kind of like "Favourite Stories"..?

I don't think my Favorite Stories page is a good example. Perhaps I don't use that feature well, because I have 16 pages of favorite stories with no way to personally organize them. I suppose I press 'favorite' on a story more as a reward to the author, instead of as a way for me to keep track. I never visit my 'Favorite Stories' page.

Here is what I envision:
In our profile, under 'media', we have 2 sections: 'Received Invites' and 'Albums'.
In each case, there are thumbnail pics with links or addresses for each actual media file.
If there were a third section 'Bookmarked', then it would display on our profile like the other two media types, and the internal storage would be a list of addresses to the media item that are held by someone else. You already do something similar under 'Interactions', to show what someone 'liked' or commented on.
It seems like it would work the same way as 'interactions', and a link to a public or a shared private item would connect to the stored media in both instances. If the item is private and the 'sharing' has expired, then the link would not work, just like with Interaction thumbnails. The user could then just delete that bookmark, or you could proactively remove it.

This could reduce the number of duplicate media items you store on your servers, because users sometimes download then re-upload media so that they can organize it themselves.

So that I understand this, the only benefit is disk space?

Well.... disk/cloud space for you is your added benefit.
For me, the big benefit is to be able to organize my access to other users' media that I would rather not just pinch and store again.