Like the Ptolemies or Seleucids and other Hellenistic successor kingdoms, or the various incarnations of the Persian empire, or any other ancient example you can think of.

I know that the above kingdoms named their various elite regiments, like Alexander's companions or the Achaemenid Immortals, or perhaps the Seleucid Chrysaspides and Chalkaspides, but those were exceptional units. I I don't know of any kingdom or nation that had regular military units that they named, let alone numbered. When ancient historians describe armies, (e.g. Polybios in book XXXI of his histories on the Seleucids) they mostly seem to refer to groups by ethnic origin. (Nisaean horsemen, etc.) I'm not sure these historians would know of any names such groups had for themselves, though. But perhaps these were only short-standing levies which didn't get names. Do we know of any more named units?

I know the navy was very important and prestigious for the Hellenistic monarchs in particular, and I know that ships were named, but I don't know of any fleets being named. The Imperial Romans typically named them after their harbours, i.e. Classis Misenensis. Did anybody else do this?

I know that (foreign) mercenaries played a big role in various armies, (like the Persians employing Greek hoplites) but I've never heard of any being organised in named units or companies. Were they?

And on a slightly different but related note, do we know how the Roman practice of giving their legions cognomen as well as numbers started? (Legio II Augusta, Legio VI Ferrate, etc.) From the timeline and from common sense I assume it only began once units acquired a more permanent nature in the first century BC, but other than that I don't know who started it or why. I mean, the purpose of fostering unit identity is obvious enough, but do any ancient writers discuss this? Is it documented elsewhere?

Thanks to anyone indulging my curiosity!