BBEdit asks for Apple ID password on every launch
Try the following steps to resolve the issue:


Uninstall the Mac App Store version. (Start Launchpad, press and hold Option key, and click on the ⨂ button shown at the top left of the BBEdit app icon.)
Download and re-install BBEdit from official website.
Install AppCleaner app.
Uninstall the version of BBEdit installed in step #2 using AppCleaner.
Restart your Mac.
Now, reinstall BBEdit from the Mac App Store.


Now you should be able to get rid of the issue.