Wordpress - wp_editor textarea value not updating

tinyMCE <textarea> element is initially unseen by the used serialize function:

$.post(
    ajaxurl,
    $('#addtag').serialize(), function(r) {
        // Content here.
    }
});

You will need to call tinyMCE.triggerSave() to make it visible.

Below is a simple snippet that should do the trick:

jQuery('#submit').mousedown( function() {
    tinyMCE.triggerSave();
});

This in an external file, enqueued with wp_enqueue_script(); it worked for the test I've conducted.

In your edited_terms function you need to save the value and in your add_tag_form_fields you need replace your test with the saved data.

something like:

class Test{

    function __construct() {

        //do_action('add_tag_form_fields', $taxonomy);
        add_action('add_tag_form_fields', array($this, 'add_tag_form_fields'));

        //do_action("edited_terms", $term_id, $tt_id, $taxonomy);
        add_action('edited_terms', array($this, 'edited_terms'));
    }

    function add_tag_form_fields($term){

        if ( current_user_can( 'publish_posts' ) ): ?>

        <div class="form-field">
            <?php 
            $saved = get_option('termmeta_'.$term->term_id);
            $saved = (empty($saved))? 'test': $saved;
            wp_editor($saved, 'mydescription', array('textarea_name' => 'my_description')); ?>
        </div>

        <?php
    }

    function edited_terms($term_id){
        if (isset($_POST['mydescription'])){
            update_option('termmeta_'.$term_id,$_POST['mydescription']);
        }
    }
}
new Test();

Now if you want a much easier way of adding extra fields of all types to your tags/categories or custom taxonomy edit forms without reinventing the wheel take a look at TAX Meta Class