Even then the browser would still need to know to which websites the user could share. The user would first need to select websites and services they want to use.
In the past, desktop browsers kind of tried to provide such features with the concept of the bookmarklet [1]. The idea was that users would drag&drop a link to their bookmark toolbar to add functionality to their browser. I think it is still used by some Read-It-Later style collection services, but mostly has been replaced with browser extensions.