Adding Author Box Meta Links with Co-Authors

These are outside of the foreach loop:

$twitter  = get_the_author_meta('twitter', $author_id);
$facebook = get_the_author_meta('facebook', $author_id);
$googleplus = get_the_author_meta('googleplus', $author_id);
$linkedin = get_the_author_meta('linkedin', $author_id);
$instagram = get_the_author_meta('instagram', $author_id);
$youtube = get_the_author_meta('youtube', $author_id);
$email = get_the_author_meta('email', $author_id);

So (remove that, and) try using $coauthor->ID with get_the_author_meta() like this:

<div class="author-box-social-icons">
<?php   
        if( $email = get_the_author_meta( 'email', $coauthor->ID ) ) {
echo '<a title="Email" href="https://wordpress.stackexchange.com/questions/309813/mailto:".$email.'" id="author-social-email"></a>';
}       
        if( $instagram = get_the_author_meta( 'instagram', $coauthor->ID ) ) {
echo '<a title="Instagram" href="'.$instagram.'" id="author-social-instagram"></a>';
}       
        if( $youtube = get_the_author_meta( 'youtube', $coauthor->ID ) ) {
echo '<a title="YouTube" href="'.$youtube.'" id="author-social-youtube"></a>';
}       
        if( $linkedin = get_the_author_meta( 'linkedin', $coauthor->ID ) ) {
echo '<a title="Follow me on LinkedIn" href="'.$linkedin.'" id="author-social-linkedin"></a>';
}               
        if( $googleplus = get_the_author_meta( 'googleplus', $coauthor->ID ) ) {
echo '<a title="Follow me on Google Plus" href="'.$googleplus.'" id="author-social-googleplus"></a>';
}                   
        if( $twitter = get_the_author_meta( 'twitter', $coauthor->ID ) ) {
echo '<a title="Follow me on Twitter" href="https://twitter.com/'.$twitter.'" id="author-social-twitter"></a>';
}
        if( $facebook = get_the_author_meta( 'facebook', $coauthor->ID ) ) {
echo '<a title="Follow me on Facebook" href="'.$facebook.'" id="author-social-facebook"></a>';
}       

?>
</div>