Rumors are, that each of these have some purpose, not discovered yet by mortal minds... and perhaps, it never will be.

Player will always start with this item to the right of their hotbar when creating a new world.

When worn in the Charm slot, it will give the wearer +2 Armor Points and +1.5 Base Attack Damage.

Every such amulet has an inscription on it, containing the name of the one who initially possessed it (except if spawned from creative/commands, or obtained through some other unintended ways). However, the inscription doesn't seem to affect it's functioning in any way - you can wear other player's Enigmatic Amulet just fine, receiving the identical effects.

There is no other way of obtaining this item, and since every player only gains one, you might want to avoid loosing it at all costs.