Earlier quoted context omitted.
You need both: the map data (OSM project), and software for viewing/using it. Ideally any app using OSM data would enable contributing to the underlaying map data. But that's probably not how it works. For what it's worth: I like Organic Maps for being more lightweight, quicker rendering & simpler configuration than OSMand. But it (still? haven't used in a while) does lack some useful features like points of interest…
> Would be nice if it were easy to share (offline) map data between apps. Download in app A, backup on sd card , use from app B, C, D, or on other device by swap/copy sd card. Maybe it's possible, but I haven't figured out how (on Android). I'm also really hoping for that. Some kind of local OSM map server that all apps in the ecosystem call to provide geodata. I run OSMand, StreetComplete, Organic Maps and Magic Ear…
Also, mobile apps often have strict privacy lately around what files they can access: they're not just sitting on the filesystem, they're in access-controlled app-specific folders. That's good for privacy/security, but a dealbreaker for first-class sharing of information between apps.