I think people are drawn to the (correct) notion that “on” can be used as an adverb, but I’m not convinced that’s the case here. If it’s part of a phrasal verb, it can be either a particle (which behaves like an adverb) or a preposition.
True, but here “hang on to” is present, not just “hang on” – “hang on” and “hang on to/hang onto” are different phrasal verbs with different meanings, and the word “on” can have different functional uses between them.
Phrasal verbs can be a verb plus particle, verb plus preposition, or verb plus particle plus preposition, but a preposition can be more than one word. So it still depends on the meaning of the phrase, which (like a lot of songs) isn’t exactly transparent here.
The line in the song that the title is taken from is “If you think we’re gonna make it, you better hang on to yourself.”
Definition 2 refers to a physical act. It doesn’t really make sense in this context. “If you think we’re going to have sex, you should dangle from yourself.” (And I really question whether that example even correctly labeled as phrasal verb, since it’s it’s a literal rather than idiomatic use of the word “hang”: A small child holding her mother’s arm for support is literally, not figuratively, hanging from it, in the same way a painting hangs from a nail.)
Even taken somewhat idiomatically to mean “cling” as in ‘target with affection’, I’m still not convinced “on” would be an adverb or particle rather than a preposition, because it’s doesn’t imply continuing or resuming an act (like “on” does when it’s an adverb), “hang” doesn’t require the word “on” to mean “cling”, and “to” can be dropped and the phrase maintains essentially the same meaning. “Hang on to yourself” can be interpreted as a snarky way of saying “take your sexual expectations off me and put them on yourself”.
Other somewhat fitting definitions (especially for the first version of the song) are here and here, which gives the meaning as more like preserve or retain. These can be more metaphorical, like “hang onto your dignity”. “Onto” is consistently labeled as a preposition, and the title is given that way in about a third of the Ziggy Stardust releases on Discogs, and on the original 7" release of this song by the Alan Corns. (Granted, since it’s four letters, it’s always capitalized here.)
The Blondie title is just “Hanging on the Telephone”, and in that case, “on” is definitely a preposition, and “hanging” is a synonym for “waiting” that doesn’t require an adverb, (although it could theoretically be “Hanging On on the the Telephone”). It’s the perfect example of definition 4 above.
