Ajax request return field without name attribute
The attribute name is missing because you have replaced the select element in ajax callback. You just need to update the #options

function insttance_ajax_fill_content($form, $form_state)
{
    $i = $form['count_field']['#value'];
    $content = getAllContentByType($form_state['values']['export_what']);

    $form['export_data']['#options'] = $content;

    return $form['export_data'];
}