If I understood your question, Yes, Flutter can do it. But that are plugins or softwares that is running in TaskBar?

Maybe you are searching it: https://github.com/leanflutter/tray_manager

You can create a New Project and in Project Type select to Plugin/Package/Module. After select in Plataform your target, Windows.

You can find more here https://docs.flutter.dev/packages-and-plugins/developing-packages