How do I get the vocabulary of a term inside an "article" type content type?
Adding the answer now. My final code in the field--entity-reference.html.twig file:

{% for item in items %}
  {% set mylabel %}
    {{ item.content }}
  {% endset %}
  {% set myclass %}
    {{ item.content['#options'].entity.vid.0.value['target_id'] }}
  {% endset %}
  {% set myclass = myclass|replace({'_':'-'}) %}
    &lt;div{{ item.attributes.addClass('taxonomy--item') }}&gt;
      &lt;a class="btn-small btn-primary tag-{{ myclass|trim }}" href="{{ item.content['#url'] }}" role="button"&gt;{{ mylabel|striptags }}&lt;/a&gt;
    &lt;/div&gt;
{% endfor %}


HERE is the code needed in a node to access the parent vocabulary of the taxonomy terms in the node. (That is, individual tags on a content type node).

item.content['#options'].entity.vid.0.value['target_id']


NOTE: This is on Drupal 8.5.3 and none of my "tags" have more than one parent.