I'm not actually familiar with modern American unit nicknames, but it does seem somewhat similar.

During the Principate, legions typically had a name and a number. This number was assigned when the legion was raised and was never changed, and wasn't entirely logical. This means you had several 10th legions, and a whole lot of 1st legions.

Most of these legions acquired a name and/or various titles as well, which could be based on:

• Their origin, e.g. Legio XIII Gemina, ('twins') a unit formed by combining two other
• The emperor who raised them, e.g. Legio II Augusta, or Legio II Traiana Fortis ('strong')
• Their place of origin, e.g. Legio I Italica, or their destination, e.g. Legio II Parthica, which was raised for Severus' invasion of Parthia.
• Stuff that makes them sound tough, e.g. Legio XII Fulminata ('lightning throwers' or 'thunderers') or Legio VI Ferrata ('Ironsides' or 'iron clad')
• Their exploits, e.g. Legio XIII Gemina again, which was awarded the titles Martia Victrix ('Mars' own, Victorious) for their role in defeating Boudicca's rebellion. Those exploits needn't have been warlike, either: Legio VII Macedonica was renamed Claudia pia fidelis ('Claudius' own, pious and faithful) for not joining their commander in a coup.

Some of these names and numbers had their origins in the final years of the republic, for example with the legions who served Caesar. The practice can't have been much older than that, though, since in Republican times legions would only serve for specific wars and campaigns and would then be disbanded, which precluded the formation of strong and lasting unit identities.

All evidence suggests that in Imperial times, Roman legionaries took great pride in their unit's history and designations, and were disdainful of other units. Soldiers who left inscriptions would usually mention their legion, for example.

If you want to know more, check out this list on livius.org, which not only has all known legions of the Principate and some later units on it, but also includes short (and sometimes not so short) histories of each of these units if you click on their names.

Source:

Adrian Goldsworthy, The Complete Roman Army (2003)