If you are using iPhone SMS Backup in your Mac with Mojavo (10.14) MacOS kindly follow below additional step to save messages from your iPhone.
Below permission is required due to new restriction introduced in MacOS.
—> Open System Preferences
—> Privacy & Security
—> Privacy ->Click on Lock Icon at bottom and Unlock -> Scroll Down left list and Select ‘Full Disk Access‘ and Click ‘+’ button the right side and select “SMSBackupApp” App from the popup dialog.
Close System Preferences & reopen “SMSBackupApp” from Applications folder.
Note: This step is not required for MacOS users running below 10.14 and Windows PC users.