Part of the problem is AbsoluteLayoutFlags you have set on the listview.
when you set it to all, you are telling the layout to start at 0,0 and go all the way to 1,1. Which is why the listview is appearing on the searchbar.
<AbsoluteLayout VerticalOptions="FillAndExpand" HorizontalOptions="FillAndExpand">
<SearchBar AbsoluteLayout.LayoutBounds="0,0,1,40"
AbsoluteLayout.LayoutFlags="WidthProportional,PositionProportional"/>
<ListView
AbsoluteLayout.LayoutFlags="XProportional,SizeProportional"
AbsoluteLayout.LayoutBounds="0,40,1,1">
<ListView.ItemsSource>
<x:Array Type="{x:Type x:String}">
<x:String>mono</x:String>
<x:String>monodroid</x:String>
<x:String>monotouch</x:String>
<x:String>monorail</x:String>
<x:String>monodevelop</x:String>
<x:String>monotone</x:String>
<x:String>monopoly</x:String>
<x:String>monomodal</x:String>
<x:String>mononucleosis</x:String>
</x:Array>
</ListView.ItemsSource>
</ListView>
</AbsoluteLayout>
Xamarin Documentation
I am sure you were referencing the documentation. I linked it here and quoting part of it. Hopefully it helps.
Proportional values define a relationship between a layout and a view.
This relationship defines a child view's position or scale value as a
proportion of the corresponding value of the parent layout. These
values are expressed as doubles with values between 0 and 1.
Proportional values are used to position and size views within the
layout. So, when a view's width is set as a proportion, the resultant
width value is the proportion multiplied by the AbsoluteLayout's
width. For example, with an AbsoluteLayout of width 500 and a view set
to have a proportional width of .5, the rendered width of the view
will be 250 (500 x .5).