You haven’t accounted for the case that there are no tags assigned in your declaration of $tag_related_posts. Your get_posts function always returns results in your line of code:
$tag_related_posts = get_posts('exclude=" . $post->ID . "&numberposts=" . $max_articles . "&tag=' . $tags_string);
If the $tags_string is null, it just returns a random list of posts and always fulfills the max post count, so your conditional to pull in by category never fires.
If you move your closing } for if($article_tags) from immediately after the foreach loop to after the if($tag_related_posts) block, your desired functionality appears it will kick in:
<?php
$max_articles = 4; // How many articles to display
echo ' <div id="carousel-related-posts" class="owl-carousel owl-theme relatedposts">';
$cnt = 0;
// Show articles based on the same tag
$article_tags = get_the_tags();
$tags_string = '';
if ($article_tags) {
foreach ($article_tags as $article_tag) {
$tags_string .= $article_tag->slug . ',';
}
$tag_related_posts = get_posts('exclude=" . $post->ID . "&numberposts=" . $max_articles . "&tag=' . $tags_string);
if ($tag_related_posts) {
foreach ($tag_related_posts as $related_post) {
$cnt++;
$backgroundImg = wp_get_attachment_image_src( get_post_thumbnail_id($related_post->ID), 'medium_large' );?>
<div class="header-wrap" style="background: url('<?php echo $backgroundImg[0]; ?>') no-repeat 50% 30%; background-size: cover;">
<header class="entry-header">
<div class="image-title">
<?php incomplete_cat_list($related_post->ID); ?>
<a class="post-title-link" href="https://wordpress.stackexchange.com/questions/288822/<?php the_permalink($related_post->ID); ?>">
<?php print get_the_title($related_post->ID); ?>
</a>
</div>
</header>
</div>
<?php } //end of foreach($article_tags as $article_tag)
} //end of if($tag_related_posts)
} //end of if($article_tags)
// Only if there's not enough tag related articles,
// add some from the same category
if ($cnt < $max_articles) {
$article_categories = get_the_category($post->ID);
$category_string = '';
foreach($article_categories as $category) {
$category_string .= $category->cat_ID . ',';
}
$cat_related_posts = get_posts('exclude=" . $post->ID . "&numberposts=" . $max_articles . "&category=' . $category_string);
if ($cat_related_posts) {
foreach ($cat_related_posts as $related_post) {
$cnt++;
if ($cnt > $max_articles) break;
$backgroundImg = wp_get_attachment_image_src( get_post_thumbnail_id($related_post->ID), 'medium_large' );?>
<div class="header-wrap" style="background: url('<?php echo $backgroundImg[0]; ?>') no-repeat 50% 30%; background-size: cover;">
<header class="entry-header">
<div class="image-title">
<?php incomplete_cat_list($related_post->ID); ?>
<a class="post-title-link" href="https://wordpress.stackexchange.com/questions/288822/<?php the_permalink($related_post->ID); ?>">
<?php print get_the_title($related_post->ID); ?>
</a>
</div>
</header>
</div>
<?php }
}
}
echo '</div>';
?>