In general, juce::Combobox consists of a juce::Label and a juce::PopupMenu. There are several ways to show a popup menu either as a modal window or not. The main problem with combobox is that although the popup is NOT shown as modal the parent component is not set up explicitely and as a result it is a nullptr. When the parent is null then popupmenu is a new window on desktop (NOT a modal window!) because addToDesktop is called. However, if the parent component is not null then the popup menu

To sum up in order to solve this problem create a Component (Label, Button or whatever suits you best) and when its pressed create a juce::PopupMenu component and set PluginEditor its parent component (I wouldn't suggest to use the parent of the target component because sometimes, depending on the parent bounds, the popupmenu might not be visible).

void MenuTestAudioProcessorEditor::showPopupMenu()
{
    PopupMenu menu;
    menu.addItem(1, "Menu Text1");
    menu.addItem(2, "Menu Text2");

    // with `showMenuAsync` the popup won't be a modal window
    // also when parent component is NOT nullptr then if menu is shown and the user moves the plugin window,
    // the menu will still be visible but inside the plugin window.
    menu.showMenuAsync (PopupMenu::Options().withParentComponent(this)
                                            .withTargetComponent (&buttonTriggerMenu)
                                            .withItemThatMustBeVisible (1) // change this depending on the current menu state
                                            .withMinimumWidth (buttonTriggerMenu.getWidth())
                                            .withMaximumNumColumns (1)
                                            .withStandardItemHeight (buttonTriggerMenu.getHeight()),
                        ModalCallbackFunction::create([this](int result){this->menuCallback(result);}));
}