Based on your previous posts I assume that you use Windows as your server OS. I am sadly unable to help you with OS-level issues on Windows. I am also unfamiliar with the quirks of the Synology OS.
You can check if your *arr-apps run as the same user as the permissions on the share (on the *arr-server). The folder and contained files of the share on the Synology NAS should have read, write and execute (rwx) permissions for the user of the share.