If I had to hazard a guess, the dancers only work on ally characters regardless of the dancer/bard's allegiance when manually controlled. Thus, when attempting to use a ring on an enemy's ally, nothing happens. Attempting to dance a greyed out ally as an NPC or enemy might work.
Using the enemy control action replay in Shadow Dragon allows you to control the enemy healers, but they cannot heal enemies, only the player units when controlled. The coding there and here might have some resemblance, but I would have to test and see if enemy/NPC dancers can only use a ring on ally units.
Unless the rings are hard-coded to specific characters/ally units, then my above statements would be incorrect.
Just tested it out, and while enemy/NPC dancers can dance greyed out units of their own faction, they cannot use rings on their respective allies. Instead, upon further testing, NPC/enemy dancers can only use rings on ally units, regardless of hidden status, but the option to dance regularly is not available on ally units if the dancer is an NPC/enemy.
I gave an enemy and NPC dancer Nini's Grace, then proceeded to grey out a few ally units, NPC units, and enemy units while leaving some active. Ally units are orange, enemies purple, and NPCs teal.
Controlling the NPC dancer, she could only dance using Nini's Grace on the ally cavalier but not on the NPC myrmidons or enemy cavaliers.
Controlling the enemy dancer yielded the same result, allowing the use of rings on ally units and being able to dance enemy units, but not rings on enemy units or regular dancing on ally units.
Attempting to dance an ally unit only allows you to use rings, which also applies to the NPC dancer.
There is no prompt to dance whatsoever next to the enemy cavalier as an enemy dancer despite holding Nini's Grace.
Before anyone points out the map, that is a map/chapter I use for debugging/testing which is why it has lava cave tile animations while snowing with a large block of ally units from Duo Geno in front of random houses.