At first I was thinking sobriquet, but that is not quite the same. Is there a term for the "the " part of such titles?
This is from Merriam-Webster
Did You Know?
Nowadays, "epithet" is usually used negatively, with the meaning "a derogatory word or phrase," but it wasn't always that way. "Epithet" comes to us via Latin from the Greek noun epitheton and ultimately derives from epitithenai, meaning "to put on" or "to add." In its oldest sense, an "epithet" is simply a descriptive word or phrase, especially one joined by fixed association to the name of someone or something (as in "Peter the Great" or the stock Homeric phrases "gray-eyed Athena" and "wine-dark sea"). Alternatively, epithets may be used in place of a name (as in "the Peacemaker" or "the Eternal"). These neutral meanings of "epithet" are still in use, but today the word is more often used in its negative "term of disparagement" sense.
from Oxford Dictionaries:
A name, designation or title.
I don't think there is a word for the 'the' part.