Is there a way to have the Combobox render the selected value like the Select in Vaadin Flow?

You can't use an arbitrary Renderer, because the text input is, well, a text input. As noted in the comments below the question, what you're really after is an icon in front of the value of the input, and while there's no nice API in ComboBox for this, you can frankenstein together a solution using the prefix slot of the vaadin-text-field input. I've adapted an example using the Cookbook recipe here. Note that there's an enhancement request that would make handling prefix/suffix components in ComboBox easier: https://github.com/vaadin/flow-components/issues/1594

public class AboutView extends Div {

    public AboutView() {
        ComboBox<Person> comboBox = new ComboBox<>();
        comboBox.setItems(getPersons());
        // Renderer for the drop down
        comboBox.setRenderer(new ComponentRenderer<Div, Person>(person -> {
            Div container = new Div();
            container.add(person.getIcon().create(), new Span(person.getName()));
            return container;
        }));
        // on value change: either clear the prefix slot or create a new Icon there
        comboBox.addValueChangeListener(e -> {
            Person p = e.getValue();
            if (p == null) {
                PrefixUtil.clearSlot(comboBox, "prefix");
                return;
            }
            PrefixUtil.setPrefixComponent(comboBox, p.getIcon().create());
        });
        comboBox.setItemLabelGenerator(Person::getName);
        add(comboBox);
    }

    public List<Person> getPersons() {
        List<Person> persons = new ArrayList<>();
        Person person1 = new Person("Foo", VaadinIcon.ARROW_BACKWARD);
        Person person2 = new Person("Bar", VaadinIcon.BAR_CHART);
        Person person3 = new Person("Baz", VaadinIcon.PUZZLE_PIECE);
        persons.add(person1);
        persons.add(person2);
        persons.add(person3);
        return persons;
    }

    public static class Person {
        public String getName() {
            return name;
        }

        public void setName(String name) {
            this.name = name;
        }

        private String name;

        public VaadinIcon getIcon() {
            return icon;
        }

        public void setIcon(VaadinIcon icon) {
            this.icon = icon;
        }

        private VaadinIcon icon;

        public Person(String name, VaadinIcon icon) {
            this.name = name;
            this.icon = icon;
        }
    }

    public static class PrefixUtil {

        private static Stream<Element> getElementsInSlot(HasElement target,
                                                         String slot) {
            return target.getElement().getChildren()
                    .filter(child -> slot.equals(child.getAttribute("slot")));
        }

        public static void setPrefixComponent(Component target, Component component) {
            clearSlot(target, "prefix");

            if (component != null) {
                component.getElement().setAttribute("slot", "prefix");
                target.getElement().appendChild(component.getElement());
            }
        }

        private static void clearSlot(Component target, String slot) {
            getElementsInSlot(target, slot).collect(Collectors.toList())
                    .forEach(target.getElement()::removeChild);
        }

        private static Component getChildInSlot(HasElement target, String slot) {
            Optional<Element> element = getElementsInSlot(target, slot).findFirst();
            if (element.isPresent()) {
                return element.get().getComponent().get();
            }
            return null;
        }

        public static Component getPrefixComponent(Component target) {
            return getChildInSlot(target, "prefix");
        }
    }
}