About Inbox
What you see in the screenshot does not quite match Menu, but FrameLayoutwith a background with shadows. This is just a custom view .

You can check it using the UI Viewer
So, you can do the same trick and instead of inflating, PopupMenucreate a custom one Viewand put it where you want.
, Menu
, Android Action ( 9 ).
menu_dropdown_panel.9:

paddings ( Paint.NET : , ):

android:popupBackground :
<style name="AppTheme" parent="Theme.AppCompat.Light.DarkActionBar">
<item name="popupMenuStyle">@style/PopupMenu.Example</item>
</style>
<style name="PopupMenu.Example" parent="@style/Widget.AppCompat.Light.PopupMenu">
<item name="android:popupBackground">@drawable/menu_dropdown_panel_background</item>
</style>
:

( , PopupMenu)
PopupMenu, :
PopupMenu popup = new PopupMenu(MainActivity.this, fab);
popup.getMenuInflater().inflate(R.menu.popup_menu, popup.getMenu());
popup.show();
,