When the possessive adjective follows the noun for emphasis, the article is not used. See the bottom of part 1 in http://www.arnix.it/free-italian/italian-grammar/possessive-adjectives-in-italian.php.
Usually suo would be before the noun. Putting it after is for emphasis. It's not >his< business. See the bottom of part 1 in http://www.arnix.it/free-italian/italian-grammar/possessive-adjectives-in-italian.php.
It is neither an exclamatory expression nor an idiom. Suo can be after any noun for emphasis on suo. When suo follows the noun, the usually required article is not used. So 'Non è affare suo' is the same as 'Non è il suo affare' except the first sentence is stronger because of the emphasis on >suo<. Perhaps it would make more sense if you thought of it as 'Questo è affare mio, non tuo.'