How to save a label for an extra user meta field and to display it later?

you could also use a label parameter in your shortcode, like this:

[user_meta user_id="1" label="publications"]

with these code changes:

* Display the selected user meta data with a shortcode */
add_shortcode('user_meta', 'user_meta_shortcode_handler');

/* usage: [user_meta user_id="1" label="publications"] */
function user_meta_shortcode_handler($atts,$content=null){ 
     extract( shortcode_atts( array(
         'user_id' => '1',
         'label' => 'publications'
     ), $atts ) );          

    $output.='<h3>'.$label.'</h3>';
    $output.= wpautop(get_user_meta($user_id, 'publications', true));
    return $output;
}

EDIT:

ok, maybe you mean something like this:

add_action( 'show_user_profile', 'extra_user_profile_fields' );

/* Add Extra Fields to the User Profile */
function extra_user_profile_fields( $user ) { 
  ?>
    <table class="form-table">
    <tr>
        <th><label for="my_label"><?php _e("Publications Label"); ?></label></th>
        <td>
            <input type="text" value="<?php echo get_the_author_meta( 'my_label', $user->ID ); ?>" name="my_label" id="my_label"/>
        </td>
    </tr>
    <tr>
        <th><label for="my_publications"><?php _e("Publications"); ?></label></th>
        <td>
            <textarea rows="10" cols="450" name="my_publications" id="my_publications"  class="regular-text" /><?php echo get_the_author_meta( 'my_publications', $user->ID ); ?></textarea>
        </td>
    </tr>
    </td>
    </table>
  <?php 
}

add_action( 'personal_options_update', 'save_extra_user_profile_fields' );
add_action( 'edit_user_profile_update', 'save_extra_user_profile_fields' );

/* Save extra user profile fields */
function save_extra_user_profile_fields( $user_id ) {
  if ( !current_user_can( 'edit_user', $user_id ) ) { return false; }
    update_user_meta( $user_id, 'my_label', esc_attr($_POST['my_label']) );
    update_user_meta( $user_id, 'my_publications', esc_attr($_POST['my_publications']) );
}

/* Display the selected user meta data with a shortcode */
add_shortcode('user_meta', 'user_meta_shortcode_handler');
/* usage: [user_meta user_id="1"] */
function user_meta_shortcode_handler($atts,$content=null){ 
     extract( shortcode_atts( array(
         'user_id' => '1',
     ), $atts ) );          

    $output.='<h3>';
    $output.= wpautop(get_user_meta($user_id, 'my_label', true));
    $output.='</h3>';
    $output.= wpautop(get_user_meta($user_id, 'my_publications', true));
    return $output;
}