why does the popup shadows do not show?

Set AllowsTransparency="True" on your Popup:

<Popup PlacementTarget="{Binding ElementName=txtName}" Placement="Bottom" x:Name="AlertPopup" IsOpen="true" AllowsTransparency="True">
    <Border Margin="0,0,15,15" Background="#272C30" BorderBrush="#6C757D" BorderThickness="2" CornerRadius="10">
        <Border.Effect>
            <DropShadowEffect Color="Black" BlurRadius="25" Direction="315" ShadowDepth="15" Opacity="0.3"/>
        </Border.Effect>
        <TextBlock Padding="7" Foreground="#ADB5BD" Background="Transparent" FontSize="12"
    Text="The file name can't contain any of the following characters:&#x0a;\ / : * ? &quot; &gt; &lt; |"/>
    </Border>
</Popup>

This is what it will look like when you run it: