Are Android system widgets within app configurable?

Volume dialog

For the volume, it is very easy to replace the volume dialog with a custom one, from android-hide-volume-change-bar-from-device you can replace it with something simple such as a progressbar in your activity, because Android provide an interface for it.

Notification dialog

You can also (if you are very motivated) have a custom display for the incoming notifications. You will need to have a service implementing NotificationListenerService running, it will have to intercepts the notifications, and send them to the main activity to display them the way you want.

Other dialogs

Others, such as the power menu will not be changeable for obvious reasons.

Conclusion

You cannot modify the style of the android system without root permission, because it is encoded in the system app.

Android offers you different way to change its behavior but it has its limitations.

Here is an example for the volume:

main_activity.xml

<?xml version="1.0" encoding="utf-8"?>
<androidx.constraintlayout.widget.ConstraintLayout xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:app="http://schemas.android.com/apk/res-auto"
    xmlns:tools="http://schemas.android.com/tools"
    android:layout_width="match_parent"
    android:layout_height="match_parent"
    tools:context=".MainActivity">

    <ProgressBar
        android:id="@+id/progress_horizontal"
        android:layout_width="match_parent"
        android:layout_height="wrap_content"
        app:layout_constraintTop_toTopOf="parent"
        style="@style/Widget.AppCompat.ProgressBar.Horizontal"/>

</androidx.constraintlayout.widget.ConstraintLayout> 

MainActivity.java

public class MainActivity extends AppCompatActivity {

    AudioManager manager;
    ProgressBar progressBar;

    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_main);

        manager = (AudioManager) getSystemService(Context.AUDIO_SERVICE);
        progressBar = findViewById(R.id.progress_horizontal);
        updateVolume();
    }

    @Override
    public boolean onKeyDown(int keyCode, KeyEvent event) {
        switch (event.getKeyCode()) {
            case KeyEvent.KEYCODE_VOLUME_UP:
                updateVolume();
                manager.adjustStreamVolume(AudioManager.STREAM_MUSIC,
                        AudioManager.ADJUST_RAISE,
                        AudioManager.FLAG_REMOVE_SOUND_AND_VIBRATE);
                return true;
            case KeyEvent.KEYCODE_VOLUME_DOWN:
                updateVolume();
                manager.adjustStreamVolume(AudioManager.STREAM_MUSIC,
                        AudioManager.ADJUST_LOWER,
                        AudioManager.FLAG_REMOVE_SOUND_AND_VIBRATE);
                return true;

            default:
                return super.onKeyDown(keyCode, event);
        }
    }

    private void updateVolume()
    {
        int currentVolume = manager.getStreamVolume(AudioManager.STREAM_MUSIC);
        int maxVolume = manager.getStreamMaxVolume(AudioManager.STREAM_MUSIC);

        progressBar.setProgress((int) ((float) currentVolume/maxVolume*100));
    }
}

Edit

Some people said the original volume bar is still visible. I directly tried the code I provided on my S10 Android 10 (Samsung framework) and the default volume bar is not visible. Therefore I also tried it on my S4 running LineageOS 10, and same result. No volume bar appear.

About the adjustStreamVolume or adjustVolume they both allow you to change volume channel but the adjustStreamVolume allow you to change a specific one for example, media, call etc. In my example it is only editing the media.