I can't seem to reproduce this. Can you please try to find simplest way to reproduce this issue? Or provide a XAML file in which this bug occurs.

Yes, it works properly when they don't have a name. If the user renames them doubleclicking in the navigator panel element's name, and then copy pastes, the bug arises. I'm attaching a xaml where this behavior happens.

Try this one

<Page xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
      xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
      xmlns:d="http://schemas.microsoft.com/expression/blend/2008"
      xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006"
      mc:Ignorable="d"
      Background="#FF4D5052">
  <Grid>
    <Rectangle x:Name="CopyPasteRectangle" Fill="#FFF4F4F5" Stroke="#FF000000" VerticalAlignment="Center" Height="100" Width="100" Margin="0,66.4,359.9,0" HorizontalAlignment="Right"/>
    <Grid x:Name="CopyPasteGrid" VerticalAlignment="Center" Height="100" Width="100" Margin="15.2,80.8,0,0" HorizontalAlignment="Center">
      <Border BorderThickness="1" Background="#FF00B5FF"/>
    </Grid>
  </Grid>
</Page>

Sorry, I somehow missed the fact that the element must have a name.
It should be fixed with r12898 -- Studio now searches for "x:Name" in the contents of the XAML before actually pasting it and replaces it's value with a name that isn't taken.
Duplicate (ALT+Drag or right click->Duplicate) now also handles the x:Name correctly (finds a non-taken name and stores it as "x:Name" rather than just "Name").