How do I get the current tag out of a taxonomy?

What you are looking for is get_the_terms(). You can get custom terms for the current post by using the following code:

$post_tags = get_the_terms(get_the_ID(), 'portfolio_tags_client'); 
if ($post_tags) { ?>
    <div class="tags-div">
        <h3><?php _e( 'Tags', 'text-domain' ); ?></h3>  
        <div class="post-tags"><?php
            foreach($post_tags as $tag) {
                echo '<a href="'.get_tag_link($tag->term_id).'" title="'.$tag->name.'">'. $tag->name .'</a>'; 
            } ?>
        </div>      
    </div><?php 
}

This should be used in your single.php or the template that renders the content of your post. get_the_term() itself can be used anywhere, but since you have to pass the post’s ID to it, you should use it in the proper template files.