If you start playonmac from the terminal command line (e.g. open terminal and type /Applications/PlayOnMac.app/contents/MacOS/playonmac) then it appears PlayOnMac will run under the permissions of 'Terminal' and you will be invited to add the permission to access removable media from terminal. If you allow this then access will be provided indirectly for PlayOnMac.
It's a workaround but not a solution when starting POM from the OSX GUI.