I’m an STM publisher, so I can say that your definition is, in fact, correct. But what does this have to do with building websites? I only say this because I don’t think you should get hung up on matters of prescriptive usage, and we both know that words change meanings in context.
As the docs explain, taxonomy has a specific meaning in the context of Hugo that I think is best explained by the movie example. The movie example, in fact, clarifies somewhat parallel versions of your animal examples.
The way that Hugo uses “taxonomies” is pretty similar to the way most generators or CMSs use taxonomies as well. Here is an article on Wikipedia that shows that Wikipedia in fact uses “taxonomies” in this way (how meta!): Taxonomy - Wikipedia
“Tags” can be a type of taxonomy. “Categories” can be a type of taxonomy. “Whatever” can be a type of taxonomy. It’s up to you to define these in your site configuration and then assign content to them in your front matter.
From the docs:
“Hugo includes support for user-defined groupings of content called taxonomies. Taxonomies are classifications that demonstrate logical relationships between content.”
Yup. Also in the docs. Think about it like this:
- You have a
tagstaxonomy defined in your configuration. - One tag you use is
cheese. You assign 7 different pages to the tagcheese by addingtags: “cheese”` to the front matter of each page. - Hugo makes it easy to build a page at
yoursite.com/tags/cheese/that lists those 7 pages.
Considerably. Look at more than one page, and give it more than a “first glance.” 
HTH.