1. iOS heavily deemphasizes the concept of a filesystem. So when I download something, what I'm really doing is "sharing" that media with another app.
2. I associate iOS's box-and-arrow icon with more than just "sharing". This player doesn't have an icon, you literally have to click text that reads "share".
It's perfectly possible the people who designed this player thought they were referencing iOS, but IMO they very much missed the mark. (Not to imply that iOS's method is particularly good UX either!)