Is Phoebus an alternate name for Apollo or is a description of Apollo? In other words, I want to know if Phoebus is a name or simply a common noun that means "light" or "brightness". There was also a Greek Titaness named Phoebe who was also related to "light" in some way or another, and her name is similar to Phoebus. That would suggests that Phoebus or Phoebe are not really proper names, but words that refer to "light".
It is used in both these ways.
Among the Ancient Greeks and Romans, just as in almost any other culture or language in the world, especially in the neighbourhood of the Mediterranean Basin, if a descriptive title or nickname gets enough usage, not surprisingly, it coagulates into something that, for all intents and purposes, is "a proper name."
William Smith's 1888 Dictionary of Greek and Roman Biography and Mythology says that Phoebus "occurs both as an epithet and a name of Apollo, in his capacity of god of the sun", citing passages from the Greek poet Homer and the Roman authors Virgil, Horace and Macrobius for examples.
It is labelled as a proper noun on Wiktionary as well as in An Intermediate Greek–English Lexicon (by Henry George Liddell, Robert Scott, Henry Stuart Jones & Roderick McKenzie). In Homer, the deity commonly appears as Φοῖβος Ἀπόλλων, "Phoebus Apollo," but also simply as Φοῖβος, "Phoebus".
Due to the later conflation of Apollo with the Titan god Helios, "Sun," who is the original and more unambiguously direct personification of the sun, Helios is also sometimes referred to as Phoebus.
The Titaness Phoebe was Apollo's grandmother (being the mother of his mother Leto). According to Line 8 of the play Eumenides, by Aeschylus, Apollo inherited the name Phoebus from this grandmother of his.
In Phoebe's case her name is undoubtedly a proper noun. She doesn't go by any other name; it is certainly more than a mere common descriptor noun, even though, ultimately, this Titaness may simply be a personification of radiance. (Indeed her only role in the mythology is as an ancestral figure to a few other deities more important than herself.)
Apollo is not typically portrayed a "warm, fuzzy" character, but as uncompromising, like truth, for which he is a patron. His brightness is fearsome, deadly, and unerring, like his arrows.
Even the inspiration he gives his priestesses, the Mantises, is a form of μανία (mania/madness).
So it's a name and a description, but a description that has linguistic (morphological) similarity to the word for fear.