skip headings

There is not h2 or h3 tag in the tooltip template.

https://git.drupalcode.org/project/tooltip_taxonomy/-/blob/2.1.1/templat...

I don't know where the tags you mentioned come from, maybe come from your theme?

Also there is a field setting to specify what tags are allowed for the tooltip.

I think this is referring to the tooltip inheriting the formatting of the text in which the taxonomy term is present. For example if the node text is bold (ie formatted as <strong>) and contains the term my term:
A sentence with my term in it.

then the tooltip text for 'my term' will be bold as well. Likewise for h2 and other heading levels.

For the <strong> example I added some CSS resets (also resets some other styles):

.tx-tooltip .tx-tooltip-text {
  font-style: normal;
  font-weight: normal;
  letter-spacing: normal;
  line-break: auto;
  text-decoration: none;
  text-shadow: none;
  text-transform: none;
  white-space: normal;
  word-break: normal;
  word-spacing: normal;
  word-wrap: normal;
}