Perhaps they actually think of themselves as Greek
Perhaps they do. A fascinating read on Christian IX of Denmark's perception of the Greek crown as thorny when it was offered to the Danes:
History is not entirely clear on why Great Britain chose Prince William of Denmark as its candidate for Greek kingship, but he was a good if not particularly obvious choice. At seventeen, he was a naval cadet at the Danish Naval Academy. As well as being young, Prince William was cheerful, charming, and exuberant, and his light personality did indeed help him weather the problems of ruling such a difficult country as Greece.
King Frederick VII of Denmark was in favor of the election, but Prince William’s father, Prince Christian, was worried about his son becoming king in such a volatile nation. So Prince Christian, who became King Christian IX of Denmark in late 1863, insisted on several conditions to make the Greek “crown of thorns” safer. He insisted that the protecting powers promise Prince William £25,000 a year if he were ever deposed. Great Britain also had to promise to give Greece the Ionian Islands, which Greece had long hoped it would do if it accepted the British candidate. And thirdly, Bavaria had to support Prince William’s election, because Prince Christian didn’t want any problems with the former king of Greece’s home country. When these conditions were settled in May of 1863, Prince Christian allowed his son to accept the throne of Greece.