How to display another field of or a simple link to the taxonomy term in the tooltip

Actually, this is already possible by editing/overriding the provided twig template at /templates/tooltip-taxonomy.html.twig, like:

<span class="tx-tooltip" tabindex="0">
  {{ term_name }}
  <span class="tx-tooltip-text">
     {{ description }}
     <span class="your-custom-class">
        {{ entity.your_field }}
     </span>
  </span>
</span>

Don't forget to add your custom class or html element in the allowed tags of your text formats!

Actually I was wrong, maybe a field I tested was displayed because of some other module or configuration giving me that token, but now I can't reproduce 😅

I can't even provide a simple link to the taxonomy term.. how to?
I see we have the twig variable "tooltip_id" but that is in the format [vocabulary_ID]-[term_ID], like "glossary-2", could we have the term id as a separate variable instead?