Typically, a lot of relatively young modern monarchies, especially ones created during the 18th century (Greece, Belgium, Romania, Bulgaria, etc.) used the style "King of the Hellenes", "King of the Romanians", etc., rather than "King of Greece", "King of Romania", because the aforementioned styles implied that they were the PEOPLE's king, that they were monarchs who ruled by popular consent, rather than king of the actual country, which typically had implications that the territory was THEIRS personally.
In the case of King Otho, he was initially an absolute monarch when he arrived in Greece, and when he was deposed and George I was put in his place, they made sure to specify that he was "King of the Hellenes" to emphasize a more democratic monarchy than the previous reign.
I can understand the confusion with the titles of the junior members of the Greek royal house. When George I became King of the Hellenes, it was agreed that all members of his dynasty would be known as "Prince/Princess of Greece and Denmark", since his father was the future King of Denmark, and all dynastic members of the Greek royal family would be male-line descendants of the Danish king. The same laws also stipulated that the crowns of Denmark and Greece would never fall to the same person, thus eliminating any member of the Greek royal family from inheriting the Danish throne, though they were still regarded as cadet members of the Danish dynasty. (Conversely, the members of the Norwegian royal house have never used the titles "Prince/Princess of Norway and Denmark", though they are alll male-line descendants of King Frederick VII of Denmark).
Nowadays, most people (i.e. the press when writing about them in shorthand periodicals or whatnot) refer to them just as "Prince/Princess of Greece", probably because they aren't aware of their descent from the Danish royal house.