Adding to @Ray Burns' original guidance: "If you don't need Names on your objects, don't name them."

There are cases where naming elements is unavoidable, but in many cases, you can avoid it. There are several ways of doing that. My examples below briefly touch on that.

Suppose you have a UserControl with a Grid containing two rows. Row 0 contains a TextBlock with some text and Row 1 contains An Ellipse. Say you want to animate both elements so that when the user hovers over the control, the text and ellipse are highlighted in orange. Now do you need to assign names to the elements? In this case, no. You can simply use binding to achieve the effect, as shown below:

<UserControl x:Class="test.Components.AnimatedTextEllipseControl"
             xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
             xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
             Background="White">

    <Grid>
        <Grid.RowDefinitions>
            <RowDefinition Height="Auto" />
            <RowDefinition Height="*" />
        </Grid.RowDefinitions>

        <!-- TextBlock with Foreground bound to the UserControl's Foreground -->
        <TextBlock Text="Sample Text" 
                   Foreground="{Binding Foreground, RelativeSource={RelativeSource AncestorType=UserControl}}"
                   HorizontalAlignment="Center" VerticalAlignment="Center" />

        <!-- Ellipse with Fill bound to the UserControl's Foreground -->
        <Ellipse Grid.Row="1" 
                 Fill="{Binding Foreground, RelativeSource={RelativeSource AncestorType=UserControl}}" 
                 Width="50" Height="50"
                 HorizontalAlignment="Center" VerticalAlignment="Center"/>
    </Grid>

    <UserControl.Triggers>
        <EventTrigger RoutedEvent="MouseEnter">
            <BeginStoryboard>
                <Storyboard>
                    <!-- Color animation from Blue to Orange -->
                    <ColorAnimation Storyboard.TargetProperty="(UserControl.Foreground).(SolidColorBrush.Color)"
                                    To="Orange" Duration="0:0:0.3"/>
                </Storyboard>
            </BeginStoryboard>
        </EventTrigger>
        <EventTrigger RoutedEvent="MouseLeave">
            <BeginStoryboard>
                <Storyboard>
                    <!-- Color animation from Orange back to Blue -->
                    <ColorAnimation Storyboard.TargetProperty="(UserControl.Foreground).(SolidColorBrush.Color)"
                                    To="Blue" Duration="0:0:0.3"/>
                </Storyboard>
            </BeginStoryboard>
        </EventTrigger>
    </UserControl.Triggers>
</UserControl>

Notice how no names were assigned to elements, but animations will still work due to binding.

Suppose you now want to animate the ellipse’s size on MouseEnter. How will you achieve that without telling the Storyboard which element to target? Naming here becomes essential to target the ellipse specifically. Here’s how the code would change:

<UserControl x:Class="test.Components.AnimatedTextEllipseControl"
             xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
             xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
             Background="White">

    <Grid>
        <Grid.RowDefinitions>
            <RowDefinition Height="Auto" />
            <RowDefinition Height="*" />
        </Grid.RowDefinitions>

        <TextBlock Text="Sample Text" 
                   Foreground="{Binding Foreground, RelativeSource={RelativeSource AncestorType=UserControl}}"
                   HorizontalAlignment="Center" VerticalAlignment="Center" />

        <!-- Named Ellipse required for targeted size animations -->
        <Ellipse Name="EllipseElem" Grid.Row="1" 
                 Fill="{Binding Foreground, RelativeSource={RelativeSource AncestorType=UserControl}}" 
                 Width="50" Height="50"
                 HorizontalAlignment="Center" VerticalAlignment="Center"/>
    </Grid>

    <UserControl.Triggers>
        <EventTrigger RoutedEvent="MouseEnter">
            <BeginStoryboard>
                <Storyboard>
                    <ColorAnimation Storyboard.TargetProperty="(UserControl.Foreground).(SolidColorBrush.Color)"
                                    To="Orange" Duration="0:0:0.3"/>
                    <!-- Targeting the Ellipse using its name (EllipseElem) to increase its size on mouse enter -->
                    <DoubleAnimation Storyboard.TargetName="EllipseElem"
                                     Storyboard.TargetProperty="Width" To="100" Duration="0:0:0.3"/>
                    <DoubleAnimation Storyboard.TargetName="EllipseElem"
                                     Storyboard.TargetProperty="Height" To="100" Duration="0:0:0.3"/>
                </Storyboard>
            </BeginStoryboard>
        </EventTrigger>
        <EventTrigger RoutedEvent="MouseLeave">
            <BeginStoryboard>
                <Storyboard>
                    <ColorAnimation Storyboard.TargetProperty="(UserControl.Foreground).(SolidColorBrush.Color)"
                                    To="Blue" Duration="0:0:0.3"/>
                    <!-- Targeting the Ellipse using its name (EllipseElem) to decrease its size on mouse enter -->
                    <DoubleAnimation Storyboard.TargetName="EllipseElem"
                                     Storyboard.TargetProperty="Width" To="50" Duration="0:0:0.3"/>
                    <DoubleAnimation Storyboard.TargetName="EllipseElem"
                                     Storyboard.TargetProperty="Height" To="50" Duration="0:0:0.3"/>
                </Storyboard>
            </BeginStoryboard>
        </EventTrigger>
    </UserControl.Triggers>
</UserControl>

The example above reiterates that while naming elements is sometimes unavoidable (e.g., targeted animations), avoid over-naming. In my example, only the ellipse needed a name for animation targeting, while the TextBlock did not. Overuse of names can increase memory usage and complexity, especially in larger applications, as Ray mentioned. The examples above are also meant to emphasize on having the proper knowledge of a language and its features.

Also note that the Name property itself cannot be animated due to WPF's restrictions. As noted in the FrameworkElement documentation:

Name is one of the very few dependency properties that cannot be animated...because the name is vital for targeting an animation.