Custom dropdown list taxonomy not saved

Question

I'm currently modifying a theme, freelanceengine, and I'd to add some customs fields in profile editing page. I succeed in creating the customs fields but my changes are not saved

    <div id="region_f" class="fre-input-field">
    <?php

    ae_tax_dropdown( 'region',
    array(
    'attr'            => 'data-chosen-width="100%" data-validate_filed = "' . $validate_region . '" data-chosen-disable-search="" data-placeholder="' . __( "Sélectionner votre région", ET_DOMAIN ) . '"',
    'class'           => 'fre-chosen-single',
    'hide_empty'      => 0,
    'hierarchical'    => false,
    'id'              => 'region',
    'selected'        => $region_arr,
    'show_option_all' => __( "Sélectionner votre région", ET_DOMAIN );
 ?>
 </div>

The var is define like this :

$region        = isset( $profile->tax_input['region'][0] ) ? $profile->tax_input['region'][0]->name : '';

I'd like to save the value of the taxonomy region and show it in the Front end but I can't find how this theme does it.

Any clues ? Thank you

Edit

I tried to manipulate functions and code according to Steve North's link This is what I tried, can't save the changes.

<?php if ( function_exists( 'extra_user_profile_fields' ) || 'save_extra_user_profile_fields' ) {
  $adress = get_user_meta( $user_id, 'address', $_POST['address'] );
    if ( empty ($adress)) { ?>
<form id="adress">
  <div class="profile-employer-secure cnt-profile-hide" id="cnt-account-default" style="display: block">
    <label for="address"><?php _e( 'Adress', ET_DOMAIN ) ?></label>
    <input type="text" name="address" id="address" value="<?php echo esc_attr( get_the_author_meta( 'address', $user->ID ) ); ?>" class="regular-text" /><br />
    <span class="description"><?php _e("Please enter your address."); ?></span>
    <input type="submit" class="fre-normal-btn fre-btn" name="" style="width: 100%" value="<?php do_action('save_extra_user_profile_fields'); ?>">
  </div>
</form>
<?php }
} ?>

I tried to create but and call the function save but I guess it's not how to do it.