In more formal times in America, one was introduced to another person, as Mr. Jones to Mr. Smith, although to a large extend this has fallen out of favor, yet formal introductions are still used in judicial settings, also as a form of respect from a younger person to an elder person, and especially from a student to a teacher. Certainly also servants were not to utilize their masters first names under virtually any circumstance, and despite the servant often being older than the child she had responsibility for, the salutation for children was often Master John, or Mistress Anne; whereas the children themselves could always address the servant by her first name.
When you take a look at the royal family in England, you take it for granted that their names are Queen Elizabeth, Prince Charles, Princess Diana, and so forth, but if you take a step back, you do wonder, why is it that these royal personages are being addressed by their first name and where is there last name? In actuality, the royal family does sort of have surnames, but they are seldom used, and in the case of the English royal family, their surname, Windsor, isn't really their surname. Windsor is, in fact, a name that they chose because previously Prince Albert, who was from the House of Saxe-Coburg-Gotha, didn't sound too British, for the dynasty of Britain's royal family, so the royal family chose Windsor as the House that they were from, and subsequently used Windsor as their last name, so that from time-to-time Queen Elizabeth was known to use the name Elizabeth Windsor. For the royalty in general, their last names, whether used or not, are often referring to the territory that they rule over, as opposed to anything else, but seeing that they are the Monarchs, the choice of such a surname is at their discretion.
Still you might wonder, doesn't the use of someone's first name, even with a salutation such as Prince or Queen seem rather too personal, especially considering that this is the royal family. In point of fact, it is, that is why in actuality, the Queen is usually addressed as "Her Majesty", and not as Queen Elizabeth. In addition, the Prince is known as the "Prince of Wales," and also we have the "Duchess of Cornwall," as well as "His Royal Highness the Duke of Edinburgh," and so forth. So in actuality, the common citizen, will not be shouting out, "hey, Queen Elizabeth," as this would be considered to be in rather poor form and not the appropriate way to address the Queen.
So in actuality, the reason that we don’t often see surnames associated with royalty has a lot to do with their lack of necessity of having a surname, and when it comes to introductions, royalty will be accorded their respect, by being addressed by their royal titles, as opposed to any intimate familiarity with their actual first names. Also, as you might suspect, the lack of surnames for royalty, goes all the way back to Biblical times, when we had King David, King Solomon, and so forth, so today's royalty and their naming conventions are consistent with the way it has always been, perhaps in the faint hope that we will see today's Kings and Queens as God's sanctioned representatives here on earth.