Get node ID or URL bằng template preprocess field FIELD NAME

You should maybe preprocess you the URL in place. Via template_preprocess_field get the node ID, build the path alias, send it to Twig.

/**
 * Implements template_preprocess_field__FIELD_NAME().
 */
function MYTHEME_preprocess_field__MYFIELD(&$variables) {

  if ($variables['element']['#entity_type'] === 'node') {
    $node = $variables['element']['#object'];
    $nid = $node->id();
    $variables['myurl'] = \Drupal::service('path.alias_manager')->getAliasByPath('/node/' . $nid);
  }
}

In field.html.twig then do:

{% if myurl %}
   <a href="{{ myurl }}">Kool Keith</a>
{% endif %}
The node in a field template is the parent entity you find in element['#object'], which you can use to build the path:

{{ path('entity.node.canonical', {'node':  element['#object'].id}) }}

Without preprocessing, directly in Twig:

{# get owner entity #}
{% set owner_entity = element['#object'] %}
{# render id #}
id = {{ element['#object'].id }}
{# render raw field value #}
field foo = {{ element['#object'].field_foo.0.value }}