Why do people say the royal family have "blue blood"?

None of the above. "Blue blood" is a translation of a Spanish phrase, "sangre azul". In Spain after the Reconquista it was a big deal to be able to claim that one's ancestry included no Moorish blood, and that therefore one's skin was European-pale. Pale people's veins show blue through the skin; brown-skinned people's veins do not. Hence, pale-skinned people could claim to have "blue blood".