Legions indeed had been numbered since the republic (consular legions being I-IV) and all other emergency one's being from V to almost infinity. titles (e.g LEGIO XX VALERIA VICTRIX (the bold part)) are mostly late republican onwards)
Auxilia were also numbered, but only from the imperial time onwards, with the units title typically being named cohors+#+region
hellenistic units were named after their taxis or ile commander, as far as I know (e.g Ptelemaios' Taxis)
Assyrians went purely by ethnicity, but I doubt you needed to know that...
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