How to add tags under categories

I believe the core issue here is that tags aren't organized by categories. As far as Wordpress is concerned, tags and categories don't have a direct relationship with each other. They are simply different ways of organizing posts, where the key difference is that categories are hierarchical (there are parent categories and subcategories) and tags are not.

The easiest solution in your use case would be to change your tags to subcategories of the parent. E.g. "appetizer" is a subcategory of "recipes."

Then go to your "Permalink" settings in the WP admin. Set a custom permalink structure to /%category%/%postname%/ and hit save twice. That should flush the permalink cash.

Then, when categorizing posts, only add them to their child category (e.g. add to the "appetizer" category but NOT the "recipe" category).

This won't include the word "tag" in your URL structure, but it will place both terms in the URL like this website.com/recipes/appetizers/post-name/