I did some investigation, and find a way to change the position of the overflow popmenu by theme(code attached at the end in case you need it), but unable to change the width to match the whole screen, not sure if Android made some limitations in menu.
If you need a drop down list exact the same of your picture, spinner may not be a perfect choice either, you'd better create a new activity that overlays origin activity and customize the new activity layout, but it's not recommended considering performance. Also, you can find something in an earlier post to see if it gives you any idea.
add a new style:
<style name="OverflowMenuStyle" parent="@style/ThemeOverlay.AppCompat.Light">
<item name="overlapAnchor">false</item><!--not overlay the toolbar-->
<!--<item name="android:dropDownWidth">10dp</item> not work-->
<!--<item name="android:maxWidth">400dp</item> not work-->
<item name="android:paddingRight">0dp</item>
<!--dropDownVerticalOffset is the property you want to avoid overlay the toolbar-->
<item name="android:dropDownVerticalOffset">4dp</item>
<item name="android:dropDownHorizontalOffset">0dp</item>
<item name="android:popupBackground">#BFBFBF</item>
</style>
Android Resource layout folder's Toolbar.xml, change popupTheme to:
android:popupTheme="@style/OverflowMenuStyle"
add an item in MainTheme.Base style:
<item name="actionOverflowMenuStyle">@style/OverflowMenuStyle</item>