How to display all posts with today’s same month and day only?

You can achieve this with the date_query parameters added in version 3.7.

To modify the main query on your posts page before it is run and apply the date_query parameters, we use the pre_get_posts action:

function historical_posts_list( $query ){
    if( $query->is_home() && $query->is_main_query() ){
        $date_query = array(
            array(
                'month' => date( 'n', current_time( 'timestamp' ) ),
                'day' => date( 'j', current_time( 'timestamp' ) )
            )
        );
        $query->set( 'date_query', $date_query );
    }
}
add_action( 'pre_get_posts', 'historical_posts_list' );

If the page you want this on isn’t your posts page, you can add a custom query in your template with the same parameters, and run a separate loop:

$posts_from_today = new WP_Query( array(
    'date_query' => array(
        array(
            'month' => date( 'n', current_time( 'timestamp' ) ),
            'day' => date( 'j', current_time( 'timestamp' ) )
        ),
    ),
    'posts_per_page' => -1,
) );

if( $posts_from_today->have_posts() ){
    while( $posts_from_today->have_posts() ){
        $posts_from_today->the_post();
        the_title();
    }
}