You can check this plugin sirikit_media_intents that implements, as the name suggests, SiriKit Media Intents for your Flutter app (i.e. the user can ask Siri "Play song on my app").

You can take a look at the source code to understand how intents get passed on to the Flutter app and back. Implementing other SiriKit intents should not be very different from the INPlayMediaIntent supported by the plugin.