I cannot answer for English, but the same phenomenon happened also in French. Here it depends at least on the date at which the name was imported: roughly, names imported before the 18th century (when authors still saw no discontinuity between Latin and vernacular, and in particular the process of transferring Latin words to French was running at full speed) were adapted. Names imported afterwards were kept intact in the nominative form.
Note that I wrote “adapted” and not ”shortened”: in general, Latin words are adapted to French by taking the accusative and dropping the unaccented last syllable. So names in -us lose the final (Augustus -> Auguste, but also Titus -> Tite, as in the Corneille play Tite et Bérénice), but for example names in -o becomes -on (Cicéron, Scipion, Bilbon^^oh ^^wait ).
However, this is only a trend and in no way a hard rule: for example, browsing through the works of Pierre Corneille, I find a lot of Tite, Martian, Crispe, Othon, but he also kept untouched a few names whose Gallicization would be weird, or conflicting with other names: Celsus, Atticus, Vinius, Lelius, Heraclius.
(By the way - just last night, an historian on French TV shortened Sejanus to Sejan, which surprised me a lot since this is very usual now, but I got the joke when Macro was also gallicized).
As a tangent, if that's ok, why does English use Leonardo and Michelangelo but not Raffaello? Is it related to /u/CubicZircon's answer about when the name was imported?
Doing this quickly on the bus so sorry for no citation, but the -ianus in names like Octavianus and Justinianus (Justinian) were added to their original names after they were adopted, showing which family they had come from. In Octavian's case, he was just Gaius Octavius until being adopted by Caesar, making him Gaius Julius Caesar Octavianus. In Justinian's case he was adopted by his uncle, the Emperor Justin I, so the same thing applied. I'd imagine after dealing with so many -ianus'es historians developed the shorthand to show they were talking about the one who had been adopted into the more powerful family.
EDIT: more directly to your point, you can call him Octavian and not lose anything because the ending still shows you're talking about him after his adoption, when he stopped being Octavius. The actual family name Octavii (someone correct me here if this is incorrect) remains as the root. Longinus was the proper name without any alterations so shortening it would change the name itself.
We just talked about Anglicization of Roman names in this subreddit at the beginning of this month, so here, from /u/Yst, is the best answer from that thread:
In most cases, English does not do this. Only in a minority of cases did English assimilate (almost always from French) the practice of spelling or pronouncing a Latin name after a fashion which reflected a Vulgar Latin form (such that Fr. Auguste where Eng. Augustus and Fr. Tibère where Eng. Tiberius). In an even fewer cases, a fully Anglicised form is adopted for its own sake. But this is usually outside the strictly historical/scholarly sphere.
Broadly speaking, this question is one of those which, as they often do, seems to demand consistency and unbreakable rules of historical linguistics in a fashion which generalises across 1500 years of linguistic history. And such answers are seldom on offer.