The Joomla! on-page menus are extensions.
They are assigned from the item that creates the page.
As an example, let's see how the Knowledgebase menu is made to appear on the right of the "Computing" knowledgebase page.
First, we select the menu item for the "How do I ..." menu:
We now select the Module Assignment tab (circled in red above):
and scroll down to the menu that we want to put on the page:
We then open the menu extension by clicking on it:
and then select the "Menu Assigment" tab:
Ticking the box puts the menu on the page.
The drop-down is used to select/deselect everything below this item in the tree, but unfortunately you have to do this each time that you add a new item (i.e. there is no inheritance of a parent item to any children which are added to it).