How to display category from recent posts?

You can use get_the_category_list() to output a comma-separated list of links to categories:

<?php
    $args = array( 'numberposts' => '5' );
    $recent_posts = wp_get_recent_posts( $args );
    foreach( $recent_posts as $recent ){
        echo '<h2><a href="' . get_permalink($recent["ID"]) . '">' .   $recent["post_title"].'</a> </h2> ';
        echo '<p>' . date_i18n('d F Y', strtotime($recent['post_date'])) .'</p> ';
        echo get_the_category_list( ', ', '', $recent["ID"] );
    }
    wp_reset_query();
?>