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Caligula ("Little Boot," singular) was a nickname lovingly bestowed on the future emperor by his father, and was used by the soldiers as a term of endearment. It was never an insult.
I don't see how being nicknamed "Gallic Cloak" can be insulting. These two may not be as flattering as "Philip the Fair," "Louis the Sun King," or "Honest Abe," for that matter, but they do fall in the same category as, say, "Gentlemanly Johnny," the nickname General Burgoyne, who almost became Prime Minister of England once, was known by among friends and enemies alike.
Upvote:3
The Romans were very into nicknames, probably because their naming system had so few personal names and because they were commonly inherited in a family, so it was not that uncommon for all the men in a family to have the same name! Nicknames were just part of life and were sometimes mocking, sometimes honoring, and probably most commonly just because.
Take the late republican general, Gnaeus Pompeius (Pompey). He adopted the nickname "Magnus" -- "the great" and was probably referred to as "Magnus" by equals in informal situations. (Note, BTW, that adopting your own nickname -- and choosing "Magnus" to boot -- took serious chutzpah. He must have been an intolerable young man.) His father, also a general and a consul, was Gnaeus Pompeius Strabo. "Strabo" means "cross-eyed" and may have been friendly and may have been mocking and probably was both. (Think how adolescent boys continually mock each other.) His grandfather was just Gnaeus Pompeius, as far as we know.
This is all pretty typical!
To judge whether or not a particular nickname is meant as a compliment or an insult or just what is usually beyond our ability at this remove. But in the particular case of Caligula, given that the story of it being given to him as a little boy and not as mockery comes from hostile sources, we can probably safely assume it is true.
Note that nicknames were used among equals. Few other than family and close friends would have addressed him as other than as Gaius Julius Caesar Augustus Germanicus.
Also, note that historians have tended to give important Romans distinguishing names. All three generations of Pompeys were probably called "Gnaeus Pompeius" in formal situations, but we tend to remember them as Pompey the Great and Pompey Strabo so we can keep them straight.
The Julio-Claudian emperors with their official names:
It's all about disambiguation!