Is it possible to define a FieldFormatter as a service I need the 'language_manager' service inside a custom FieldFormatter. I'd prefer not to use \Drupal::service('language_manager').

Is there a special tag for FieldFormatters, or any other method to define it as a service?
You don't need to define your FieldFormatter as a service. FieldFormatters are instantiated inside the FormatterPluginManager's createInstance method (FormatterPluginManager::createInstance) which calls a create method on the formatter plugin object, passing a $container object as a parameter. 

But this happens only if your FieldFormatter implements the \Drupal\Core\Plugin\ContainerFactoryPluginInterface. From then on, you are able to inject the service inside your FieldFormatter.