As I thought, the Greek of the "word" koumpounophobia is modern not Classical. Because Classical Greek β was a stop /b/, but Modern Greek β is now a fricative /β/, latter now uses a digraph μπ to stand for the stop /b/ initially in a word or /mb/ elsewhere. The Modern Greek noun for button is κουμπί (koumbi, /kumbi/). Not sure where the rest of the word before -phobia comes from, but I hazard to guess that somebody looked in a (Modern) Greek-English dictionary and looked up button in the English half and settled on the verb κουμπώνω (koumpōnō. /kumbono/) 'to button'. From there they probably mixed up the omega (ω) with the digraph for /u/ (ου). Of course, not knowing much about Greek (Classical or otherwise), they left in the first person singular ending, the second omega (ω).
[Addendum: Looking around further, it seems that Neil Gaiman coined the word. Also, the Modern Greek word (which is not in the Greek Wikipedia) κουμπί is probably related to the Classical Greek κουβιον (koumbion) 'buckle'.]This message has been edited. Last edited by: zmježd,