How to display related posts by same publish date?

Get the date of current post:

$date = get_the_date('Y-m-d');

Get year, month and day from the $date variable.

$exploded = explode('-', $date);
$year = $exploded[0];
$month = $exploded[1];
$day = $exploded[2];

Lastly, query the posts that has the same publish date.

$args = array(
    'post_type' => 'post',
    'post_status' => 'publish',
    'date_query' => array(
        array(
            'year'  => $year,
            'month' => $month,
            'day'   => $day,
        ),
    ),
);
$query = new WP_Query( $args );

Add any query arguments and display the query results as you wish.