Presumably this is because they don't want hangouts to become a de facto file sharing service, by the fact that the person you're messaging could be offline and

Presumably this is because they don't want hangouts to become a de facto file sharing service, by the fact that the person you're messaging could be offline and you might expect the content to be archived/re-downloadable. The fact that pictures are shareable is (I think) because they do/can put the pictures into a google photos gallery. What you probably want to do is put it in google drive and then share a link in hangouts. (Not a googler)