Conditional concerning a selected tag not working

isset() checks to see if a variable is set, ie, has a value other than null. The way you’re using it will always return true because you’ve set $posttags to 'animali permessi', which is not null.

I think you’re looking for the WordPress function has_tag() instead.

if (has_tag( 'animali permessi', $post ) ) {
    $supplemento_animali = '<div class="suppl_animali">'. get_the_author_meta( 'supplemento_animali', $post->post_author ) .'</div>';
}