OK, I think I figured out what is going on in my org. I published a web map with the basemap service, shared it publicly and with the basemaps group that is part of the site. I then authored an app from a template that used the web map and shared it with a different group on the site (apps). Even though the web map was public, because it was shared with another group (basemaps) and not the apps group I was trying to share the application with, the app does not show up in the apps group. If I share the web map with the apps group, the app that uses it can be shared with the apps group. If I share the web map with no groups (but leave the public sharing set), it also can be shared with the apps group.
I'm not sure if that's a bug or a feature :confused: