They do not call themselves kings or princes. They bear titles that belong to their royal house.
You have to differentiate: A state of a real house, they are not the same.
The Greek royal house is a European royal house, recognized by the rest of the royal houses, they carry titles that are valid for the royal houses, and for the States with monarchy that recognize the titles of royalty. Britain allowed to be identified with the title of royalty, Princess Theodora in her British passport she is "Princess Theodora", because Britain is a state with a monarchy and recognizes the protocols of European royal houses. Princess Irene acquired the Spanish passport, Spain is a country with a monarchy and recognizes the protocols of royal houses, this allows identification with the title. Currently, King Constantine, due to a 1994 Greek law that unjustly withdraws his passport, has to carry the Danish passport, in which it appears as HM King Constantine, Denmark is a state with a monarchy that recognizes the European royal houses. The International Olympic Committee , expressly, recognizes the titles of the royal houses, King Constantine who is one of the oldest members of the committee is recognized as king, this and due to the Royal House, not the state from which they come The Republics do not recognize the value of the European royal houses and their titles, they only recognize the Heads of State and their titles, the Greek Republic is not the exception. For the Greek state in the form of a republic, no royal title is valid, because it does not recognize the royal houses, only the positions they occupy and the title that this state has given to the institution, but no royal families.
And finally, I am Greek, and I am free to believe in whatever I want, the Greek state cannot forbid me to believe, I believe in the monarchy and the titles.
Philippos is Prince, and his passport say "Prince Philippos".
Titles are created by royal houses, not states. States cannot create Royal Houses titles. The republic does not give legal effects to titles, but it does not exterminate titles, because it is not competent to do so.