What’s in a name? #11

I was trying to find out His Royal Highness The Prince Philip, Duke of Edinburgh’s real surname, the one he presumably had before he got married.

Prince Philip, Duke of EdinburghWikipedia
Prince Philip (Greek: Φίλιππος, romanized: Fílippos) of Greece and Denmark was born on the dining room table in Mon Repos, a villa on the Greek island of Corfu on 10 June 1921, the only son and fifth and final child of Prince Andrew of Greece and Denmark and Princess Alice of Battenberg. A member of the House of Glücksburg, the ruling house of Denmark, he was a prince of both Greece and Denmark by virtue of his patrilineal descent from George I of Greece and Christian IX of Denmark, and he was from birth in the line of succession to both thrones.

So he never had one, unless we call ‘of Greece and Denmark’ a surname?