Satan is never called "morning star", this epithet, used by the Prophet Isaiah, refers to the king of Babylon, not the devil. Its use by Isaiah is satirical/sarcastic--the king of Babylon, an ordinary human being, claimed to be divine, among the gods, among the stars in the heavens--but ultimately he is nothing and would amount to nothing.
Its use in reference to Jesus is sincere; the planet Venus (the "morning star") is the last "star" we see just before the sun rises, hence "morning star" or "dawn star"--it signals the dawn, brings the dawn. Which is why in Greek it is called
eosphoros ("dawn-bringer") and
phosphoros ("light-bringer"), translated into Latin as
lucifer (also meaning "light-bringer" and a reference to the planet Venus).
None of these are names, titles, or epithets given to Satan in the Bible. The reason many people think Satan's name is "Lucifer" is because of a rather peculiar interpretation of Isaiah 14 that arose in the middle ages in which Isaiah 14 was viewed as speaking, mystically and cryptically, about the fall of Satan--but this view is entirely eisegetical and has no basis in the text itself.
For most of Christian history the term "lucifer" in the Latin-speaking West was used to refer to Jesus Christ (on account of 2 Peter 1:19), and on a few occasions was a proper name given to Christians, for example there were several bishops in antiquity named Lucifer, for example the 4th century Sardinian bishop,
St. Lucifer of Caligari.
-CryptoLutheran