I see no reason to restrict this to just unarmored characters, especially given how much of a rules kludge it is.
Making a complicated ruling just to narrow the item to only a very few useful cases does not make much sense to me personally but your mileage clearly varies.
In 5E very few characters are unarmored. Not draconic sorcerer's, not monks, not wizards with Mage Armor, not Rogues in light armor...
Wildshaped druids... The paladin caught out of her plate armor... And that's about it.
If you want a rule that's kept simple and easy, you are going to get all of them. That's how 5E is set up.
No that's OK, if my understanding of what you were suggesting was correct, then I'll stand by my version. The Amulet of Natural Armour in 5e should only benefit unarmoured wearers (unless the armoured AC is lower), including those who already benefit from natural armour. So it would help barbarians, monks, wizards, sorcerers, druids in wildshape, polymorphic characters and animal companions, plus armoured characters while resting. That is still a wide number of characters. Whether it should be attuned is another matter. It's weaker than a ring of protection so I'd say no.
The simple and far less useful on is the version that gives a fixed AC. Although less useful, it's equivalent to a permanent mage hand or Bracers of Defence so would probably need to be attuned. If you want to spice it up, allow it to grant damage resistance for 1 minute once per day but it would end up being worn for that rather than the natural armour.
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