Youka's comment was the most helpful for me. Namely these two points:

1. in SERVER_MODE the PassFile path in servers.json is relative to the user's storage directory, even if the path is starting with /;
2. in the user's storage directory name @ character should be replaced with _, such that if [email protected], the directory is /var/lib/pgadmin/storage/admin_example.org/.

For whatever reason I am unable to comment directly under Youka's answer.