How do I select a tab with right mouse click event in WPF

You have to style your TabControl adding event for MouseDown then select the tab item accordingly.

This is the code:

XAML

At fist, you need to define reosurces for TabControl style and also a style for a Grid. The latter is needed because you can’t define an event handler within the TabControl style directly.

<Window x:Class="WpfApp7.MainWindow"
    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"
    xmlns:local="clr-namespace:WpfApp7"
    mc:Ignorable="d"
    Title="MainWindow" Height="450" Width="800">
<Window.Resources>
    <Style x:Key="GridStyle" TargetType="Grid">
        <EventSetter Event="Mouse.MouseDown" Handler="UIElement_OnMouseDown"/>
    </Style>

    <Style x:Key="TabcontrolStyle"  TargetType="{x:Type TabControl}">
    <Style.Resources>
        <Style TargetType="{x:Type TabItem}">
            <Setter Property="FocusVisualStyle" Value="{x:Null}" />
            <Setter Property="Background" Value="Transparent" />
            <Setter Property="Template">
                <Setter.Value>
                    <ControlTemplate TargetType="{x:Type TabItem}">
                        <Grid Height="20"
                              Background="{TemplateBinding Background}"
                              SnapsToDevicePixels="true"
                              Style="{StaticResource GridStyle}">
                            <ContentPresenter Margin="10,0"
                                              HorizontalAlignment="Center"
                                              VerticalAlignment="Center"
                                              ContentSource="Header" >
                            </ContentPresenter>
                        </Grid>
                        <ControlTemplate.Triggers>
                            <Trigger Property="IsSelected" Value="false">
                                <Setter Property="Background" Value="Transparent" />
                            </Trigger>
                            <Trigger Property="IsMouseOver" Value="true">
                                <Setter Property="Background" Value="{x:Static SystemColors.ControlBrush}" />
                            </Trigger>
                            <Trigger Property="IsSelected" Value="true">
                                <Setter Property="Background" Value="{x:Static SystemColors.ActiveCaptionBrush}"/>
                            </Trigger>
                        </ControlTemplate.Triggers>
                    </ControlTemplate>
                </Setter.Value>
            </Setter>
        </Style>
    </Style.Resources>
    <Setter Property="Template">
        <Setter.Value>
            <ControlTemplate TargetType="{x:Type TabControl}">
                <Grid KeyboardNavigation.TabNavigation="Local">
                    <Grid.RowDefinitions>
                        <RowDefinition Height="Auto" />
                        <RowDefinition Height="*" />
                    </Grid.RowDefinitions>
                        <TabPanel Name="HeaderPanel"
                                  Panel.ZIndex="1"
                                  IsItemsHost="True"
                                  KeyboardNavigation.TabIndex="1" />
                    <ContentPresenter Name="PART_SelectedContentHost"
                                      Margin="10"
                                      Grid.Row="1"
                                      ContentSource="SelectedContent" />
                </Grid>
            </ControlTemplate>
        </Setter.Value>
    </Setter>
    </Style>
    
</Window.Resources>
<Grid>
    <TabControl x:Name="MyTabControl" Style="{StaticResource TabcontrolStyle}">
        
        <TabControl.ContextMenu>
            <ContextMenu Name="tabContextMenu">
                <MenuItem Header="Change Tab Name" />
                <MenuItem Header="Save Favorite Layers" />
            </ContextMenu>
        </TabControl.ContextMenu>

        <TabItem Header="First">First Content</TabItem>
        <TabItem Header="Second">Second Content</TabItem>
        <TabItem Header="Third">Third Content</TabItem>
    </TabControl>
</Grid>

Code behind

In the code behind you can equals the TabItem header to selected the TabItem accordingly.

private void UIElement_OnMouseDown(object sender, MouseButtonEventArgs e)
    {
        ContentPresenter contentPresenter = null;

        if (e.Source is Grid)
            contentPresenter = (ContentPresenter) ((Grid) e.Source).Children[0];
        else if (e.Source is ContentPresenter)
            contentPresenter = ((ContentPresenter) e.Source);

        if (contentPresenter == null) return;

        var header = contentPresenter.Content.ToString();

        var selectedIndex = -1;
        foreach (var item in this.MyTabControl.Items)
        {
            selectedIndex++;

            var tabItem = item as TabItem;

            if (tabItem?.Header != null && tabItem.Header.ToString().Equals(header, StringComparison.InvariantCultureIgnoreCase))
            {
                this.MyTabControl.SelectedIndex = selectedIndex;
            }
        }
    }