This is a question raised by @domenic on the TAG review for the <popup>, here. Copying the relevant part from that comment:

We need to figure out what the similar semantic space, and corresponding role, is for <popup>. The explainer example seems to give several types of controls: menus, form element suggestions, content pickers, and teaching UI. What is the ARIA role for each of these? Perhaps each should have a separate element, not <popup>, which implements the appropriate semantics (and behaviors?? I can't imagine a teaching UI needs the exact same behaviors as a menu). Or perhaps some of those are better suited for <dialog> instead of <popup>.

Should we be proposing several "flavors" of popup instead as separate elements, e.g. <tooltip>, <listbox>, etc.?