How Do I Register Two Taxonomies with one Parent of the Other?

A taxonomy cannot be a parent/child of another taxonomy. The “hierarchical” notation indicates that you can have a hierarchy within that taxonomy. This answer addressed car Make & Model graphically.

Think “category” – “subcategory” relationship.

You can, however, have multiple levels of terms within a given taxonomy. If you look at the wp_insert_term() codex page, you will find a “parent” parameter that sounds like what you want to use.

So, within taxonomy “make-model” you might have Term=Ford, terms with $parent==”Ford” => (Astro,Mustang,F150), $parent==”Honda” => (Civic, Accord, Fit), etc.

This article by WP Mayor goes over custom taxonomies, and hierarchical taxonomies, in great detail.