x:name does not set the name property of a Control.

Now also Code like this:
<Window xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
        xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml">
  <Canvas>
    <Button x:Name="aa"
            Content="{Binding Name, ElementName=aa}"
            Width="75.2"
            Height="23.2"
            Canvas.Left="85.6"
            Canvas.Top="197.6" />
  </Canvas>
</Window>

works!