English idioms are a spontaneous speech production with fixed & distinctive quality to the native speakers of English. Therefore, they seem to fall under the category of 'functional language'.
If you are referring to the distinction between functional words/closed class vs. lexical /open-ended categories, idioms are not part of "functional language" as you are calling it. Functional categories include pronouns, articles, complementizers, demonstratives, possessive markers, tense/aspect, inflection, etc.
"Functional language" is a term most often used with respect to computer programming. In fact, I am not sure that I would call idioms "spontaneous." Rather, they are a kind of standardized shorthand that conveys meaning via a cultural reference that is literary, historical, or even drawn from the world of sports.
Thank you Michael for the reply! It helps a lot. I happened to see the term "functional language" in a website which provides interesting TESL activities. Attached a cropped screenshot of one of it pages. Anyway, I'm still in a dilemma.