The Spartans are said to have gone into battle with little strips of wood tied to their wrists with their name on them. It's only a very late source telling us this, but if it is correct, that might push the concept back to the 5th century BC, if not earlier.

It would make sense for the Greeks to invent something like dog tags, given the importance of recovering all fallen citizens after a battle. Most city-states relied on friends and relatives in the battle line to identify the dead, but the Spartans may well have come up with a more efficient system.