I couldn’t find a way to style (set the background color) of the filler column area that appears when the (vertical) scrollbar spawns.

I am talking about this part:

It should be possible with either CSS or code styling.
I created this MRE:

record Person(String name, int age) {
}

@Override
public void start(Stage stage) {
    TableView<Person> test = new TableView<>();
    test.setColumnResizePolicy(TableView.CONSTRAINED_RESIZE_POLICY_ALL_COLUMNS);

    // Create columns
    TableColumn<Person, String> nameColumn = new TableColumn<>("Name");
    nameColumn.setCellValueFactory(p -> new SimpleStringProperty(p.getValue().name));

    TableColumn<Person, Integer> ageColumn = new TableColumn<>("Age");
    ageColumn.setCellValueFactory(person -> new SimpleIntegerProperty(person.getValue().age).asObject());

    test.getColumns().addAll(nameColumn, ageColumn);
    ObservableList<Person> data = FXCollections.observableArrayList(
            new Person("John Doe", 30),
            new Person("Jane Doe", 28),
            new Person("Michael Smith", 35),
            new Person("Emma Johnson", 22),
            new Person("Olivia Williams", 27),
            new Person("Noah Brown", 31),
            new Person("Liam Jones", 29),
            new Person("Mason Garcia", 26),
            new Person("Lucas Miller", 24),
            new Person("Ethan Davis", 32),
            new Person("Ava Martinez", 21),
            new Person("Sophia Rodriguez", 25)
    );
    test.setItems(data);

    test.setMinHeight(500);
    stage.setScene(new Scene(test));
    stage.show();
}