The term "national" doesn't simply mean "having to do with a nation state".
I promise that I do not mean to be condescending with this, but you can verify in Merriam-Webster's entry for "nationality" which is referenced by the entry for "national", or also the Wikipedia article on "national cuisine". The Merriam-Webster entries specifically give examples showing that my usage encompasses the examples you gave.
The terms "ethnicity" and "ethnic" are not problematic per se. There are a few reasons people object to usages such as "ethnic cuisine", one of which I mentioned in the comment you are replying to.