Following from this comment: #112239 (comment)

If you wrap the newly implemented Menu widget in a ScrollbarTheme widget, the style won't apply for the Menu scrollbar. This happens because the menu is in an overlay, which does not share the context with the Menu widget. Currently, the only way to apply some style to the MenuItem scrollbar is to change it for the whole app.

A solution would be to either allow the MenuItem widget to pass in a scrollbar theme and apply it to the overlay or somehow make the Overlay context the same as the MenuItem. Not sure if any other alternative is feasible, like getting the style from the context and applying it to the Overlay?