I'm not sure where they originated, but the purpose is basically membership in the group. It allows you to identify yourself as an official member (or in many cases, ex-member) of a particular regiment, unit, etc., whether in uniform or not. It's a unit cohesion, morale thing.
I heard this from a former member of the Cdn Airborne Regt. Every member of the Regt was issued a Regimental coin, which they were supposed to carry on their person at all times. If someone made a challenge (i.e. produced their coin), all other members had to produce their coins, or pay a forfeit -- usually in the form of buying a round for the house. If everyone did indeed produce their coins, the challenger had to buy the round.
One guy used to even take his into the shower with him, just in case, secreted in his, er, tailgun, so to speak.
Thing about it is, a newbie in the 101st Airborne could wander into a pub in Podunk, whip out his coin, and be met by a grizzled old ex-Sergeant-Major who para'ed into France in '44. Instant connection.
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