How do you assign a QML Item to a component property in QML and then use that object inside the component?

There is much better solution:

/* MyObject.qml */

Rectangle {
    default property alias data /* name can be any */ : inner_space.data

    /* ... You can put other elements here ... */
    Item {
       id: inner_space

       /* ... Params ... */
    }
    /* ... You can put other elements here ... */
}

And now we can do all we want!

/* main.qml */

Rectangle {
    MyObject {
        Button {
             /* ... */
        }
    }
}

Thanks to user bobbaluba for suggesting to use the data property rather than children.