Popup (and Menu) QML Type doesn't work as intended if it is placed in a file that has an Item as a root element. 

With Popup's default settings or with next implicitly stated settings:
modal: false
closePolicy: Popup.CloseOnPressOutside | Popup.CloseOnEscape
Popup can't be closed with anything but using Escape key. Replacing Item root element with Window fixes this behavior. 

Alternative option so far is to make Popup modal and style it's background with transparent Rectangle:
modal: true

Overlay.modal: Rectangle {

color: "transparent"}