Show posts from categories instead of tags

In the function wp_get_post_terms() you can select the taxonomy. the default is post_tag you can change it to category.

And in the WP_Query args you need to change the tag__in to category__in.

// get the categories
$categories = wp_get_post_terms($post->ID, 'category');
if ($categories) {
    $categories_ids = array();
    foreach($categories as $individual_cat) $categories_ids[] = $individual_cat->term_id;
    $args=array(
        'category__in' => $categories_ids, // Select posts from this categories
        'post__not_in' => array($post->ID),
        'posts_per_page'=>10, // Number of related posts to display.
        'ignore_sticky_posts'=>1
    );

    $my_query = new wp_query( $args );
    $num=1;
    while( $my_query->have_posts() ) {
        $my_query->the_post();

        ?>
        <li>
            <a href="https://wordpress.stackexchange.com/questions/291639/<?php the_permalink(); ?>">
                ...
            </a>
        </li>
        <?php
    }
    wp_reset_postdata();
}