I'd argue that mobile apps have been trying really hard to make the "local storage" concept go away: auto-save, cloud as the canonical storage, transparently managed local caches... and the user needs not care about where their data is at any moment: it's "in the app".

From that point of view, "save to local storage" makes sense to become a "share with the app that happens to save to local storage"... but that could also be any other app which would keep the content for later use, not necessarily on local storage.

If anything, over time we might see the emergence of a "move the content" context, from one app to another, in addition to "share the content".