There's a convention to append a tilde to files/folders that are backups, so presumably some app at some point made a copy of /usr/local/share/applications, and then the original one got deleted?
That's odd. I'll delete it since it's empty and create a new one called just "applications". I tried having both yesterday, but it didn't work even though I put the Syncthing .desktop files inside and restarted my PC.