This appears to be a limitation with the MaterialDesignCardFlipper
The simplest solution would be to use the default style for the flipper (omit the Style="{StaticResource MaterialDesignCardFlipper}" if you have it), and set the BorderThickness property to 0.

However it does appear that being able to customize the shadow depth on the flipper is something that should be supported. If anyone wants to submit a pull request for it, it should simply be a template binding on the Card inside of the MaterialDesignCardFlipper style (linked above). This should allow someone to customize the depth of the shadow doing something like:

<materialDesign:Flipper Style="{StaticResource MaterialDesignCardFlipper}" VerticalAlignment="Top"
                        materialDesign:ShadowAssist.ShadowDepth="Depth0">
...
</materialDesign:Flipper>

Hi, without the style MaterialDesignCardFlipper, every content element are displayed strangly (font size are bigger for example).