How to automatically set a Template Page Name next to a page in menu screen such as WooCommerce pages, front page, or posts page in wordpress?

Use the display_post_states hook:

function custom_display_post_states( $states, $post ) {
    if ( 'Services' === $post->post_title ) {
        $post_states['custom-content'] = 'Services Page';
    }

    return $post_states;
}

add_filter( 'display_post_states', 'custom_display_post_states', 10, 2 );

or you can do by ID

if ( 1 === $post->ID) {
    $post_states['custom-content'] = 'Services Page';
}

To check if page has template:

function custom_display_post_states( $states, $post ) {
    $template = get_page_template_slug( $post->ID );
    
    if ( $template == 'YOUR_TEMPLATE_FILE' ) {
        $states['custom-content'] = 'Services Page';
    }

    return $states;
}

add_filter( 'display_post_states', 'custom_display_post_states', 10, 2 );