How do I create and apply a configuration schema for an existing custom module?
To have the "Translate" tab listed on your module's configuration page you additionally need to provide a module_name.config_translation.yml in your module's root.

module_name.settings:
  title: 'Module Name'
  base_route_name: module_name.admin_settings
  names:
    - module_name.settings


Look for existing module_name.config_translation.yml file in core and contrib and you'll get the pattern no problem.