The Key/Fae/Fang/Fish/Spider triggers likely take priority; then anyone etheric gets Eye charm (whether Inherent Etheric or Taking Eclipse); and then Human; and if none of above apply you don't get any charm.

This could be determined from techniques those opponents use or other opponent flags.
Some notes:

Key:

• Specifically tagged to the chained figure for the puzzle/boss/skill.

Fang:

• We know that there is an existing Hound flag; but Drone Statue does not qualify (though does trigger fang).
• Most of these use Ghoul Bite, but Drone Statue does not (though triggers fang); and others like Painted Woman does use Ghoul Bite (though does not trigger fang).
  • Perhaps triggers on {uses eating technique if Master Defense is eating technique; and otherwise not demihuman} but that seems unlikely.
• There is also the side-oddity that apparently if you have a Bone effect and Etheric active, the fang is temporarily unlocked as long as you have those effects active but cannot be pulled off. Can other charms be unlocked in temporary ways?

Fish:

• Seems consistant with Aquatic.
  • Deep Hunter has Ocean Eyes, but triggers on Lake Serpent as well, which doesn't.
  • To test - if writhing mass underwater unlocks it? Probably not.

Fae:

• Not sure on trigger. Mushroom Scout does not have Fae Eyes.
• Mushrooms and Fae are at odds a bit in David's quest; they are obviously inter-related in some ways but not fully aligned.

Spider:

• Probably not triggered based on what techniques enemies use (eg. Midgard SD1 only uses Ranged).
• Centipedes also trigger this, a bit of a belated surprise. (edit: Crypto says he asked in a bug report if centipedes should be tagged as "skittering" maybe, that may have been added later.)
• Notably: The Twisted Patron is Spider but Spider Patron is Eye. What makes the etheric version of this opponent spidery?

Eye:

• Seems to trigger on all humans that are Taking Eclipse, and on enemies that are "Inherently Etheric".
  • Inherent Etheric is a flag that Kinak has confirmed exists but is not known to be able to be tested for (aside from this item maybe)
  • Notably: The Design Student does not trigger Taking Eclipse, but does trigger the Eye charm. She claim she does not take Eclipse but there is something odd going on with her odd sketchbook. Possible she has the inherent etheric flag? Has she somehow internalized ethericness? Not sure what the story implications are there.
  • No way to confirm but it seems the non-green opponents listed above (Living Sludge, Tusked Shadow, Writhing Mass, Giant Maggot, Ghoul King, Navigator) are probably tagged as inherent-etheric as well.

Skull
All remaining humans that don't meet above criteria.

None
The rare opponent that doesn't meet any of above criteria; notably sappy stuff like the faceless worker/cubicles.

The eye charm is also an oddity: successfully gotten from the writhing mass, living sludge, spearwoman, etc. but NOT from the tree of hands or rat king. I wonder if that is flag based as well?

See Unlocked chain discussion for more details.

Also - no charm for killing old man mushroom.

EDIT 2 Mar 2015:
Podcast 40: this is it

0 SP = 0 copies
1 SP = 4 copies (nightmarish gang jacket only)
2 SP = 4 copies (abstract domino mask only)
3 SP = 3 copies (shining glasses only)
4 SP = 3 copies (black strands)
5 SP = 2 copies
6 SP = 2 copies
8 SP = 1 copies
9 SP = 0 copies

-2 SP = 0 copies

Now to figure out how to shatter my opponent's bonds…

(I would imagine this is checked at the start of combat… there isn't a lot that can change your stealth power mid-way through… Blood In The Snow maybe though probably can't get that unless you are etheric. Maybe some weird combination of an antique pocket mirror reflecting a gaze effect which you can then use to get Gaze Beyond effect? EDIT: yes, I can confirm that if you get +2 SP from Gaze Beyond, you still keep the same of techniques in the deck — it is only checking SP at the start of the fight.)

Midgard Scientist = skull

(The eye charm gave a stringy eyeball and Arkaim was etheric. I'm not sure if my clones were using the stringy eyeball version but I doubt it. I would suspect that part is responsive.

Should we keep a list of what we've fought with the chain?

I was planning to add a column to the opponents testing page when I had a spare second. In the meantime, here is as good as anywhere to keep a list I suppose.

It's not clear exactly how you un-unlock (lock?) charms again, other than resetting presumably so some opponents may be harder to test without resetting a bunch.

EDIT: also - nothing special for killing spectral sailors. I thought that there might be since these are ghost chains.

Edit: If my only equipment is the charmed chain and a +1 offhand like the bat paperweight or a cabling nest, I wind up with no bonus duration at all, not even that which the offhand should provide. Asking Kinak if something weird is going on, or the modifier is just less straightforward than it first appeared.