Are there any powers that are unique only to one mage? If so, why?

Yes.

Quaraun's hair seemingly possesses a magic ability, to be usable as limbs, like hands or feet. It is actually not magic at all, but Humans believe it is magic.

Quaraun is often described as "gliding a few inches off the ground, his feet not touching the floor", which leads to the belief that he levitates. In fact, what he is doing is using hiss hair instead of his feet to walk. Because Quaraun is only five feet six inches tall yet his hair is twelve feet long, he is able to use his hair to lift himself off the ground by a few inches or even several feet.

The secret to Quaraun's supposedly magic hair, is that t is not actually hair at all, because Quaraun is not actually an Elf. Quaraun is parasitic, alien JellyFish, disguised as an Elf, wearing the body of a dead Elf like a fur coat. Quaraun's hair, is actually thousands of very fine, thin JellyFish tentacles, which is why his hair is bioluminescent and glows in the dark, and why his white hair, shimmers silver and blue in the light resembling fish scales.

When around other people, Quaraun moves as an Elf does, walking on his feet, however, when alone or with people he trusts, Quaraun uses his tentacle-hair to move. Doing so lifts him up off of the ground, and most people are so busy looking at him hoovering, that they do not notice the way his hair stiffens to support his body, while long parts of his hair, the parts pooling around him on the ground, are actually stiffened into spider-like "feet" moving along the ground.

Quaraun is also able to use his hair like a weapon. When threatened, and feeling he has no other way to escape, Quaraun''s hair takes on a Medusa-like life of it's own, lashing out at his attacker, either pulling them away, or wrapping around the. When confronted by life-threatening situations, Quaraun has been known to use his hair to strangle his attacker to death.

Being a JellyFish, similar to the Portuguese Man of War, Quaraun tentacles are full of highly toxic venom and he can also sting his enemies to death.

Quaraun is usually described wearing a saree and a hijab and a veil, all at once, wearing all 3 as a way to conceal the true nature of his hair. If you don't examine it too closely or if you don't touch it, Quaraun's hair, appears to be nothing more than, long silken stands of hair. Removing the veils, shawls, saree, and hijab allows the slithering, living, movement of Quaraun's hair to be seen, thus why he rarely uncovers it.

Because Quaraun's hair is not actual hair, and is instead are his JellyFish limbs, thousands of tentacle arms and legs, Quaraun never cuts his hair and becomes deeply offended by any suggestion that he does.

In a scene where he is overpowered by his attackers and they shave his hair, we see Quaraun left bleeding to death, as his blood drains from the thousands of severed tentecles.This "cutting of his hair" leaves him in agonizing pain for months, and while, like any JellyFish he can regrow his severed "limbs" it takes 30 years for his tentacles to grow back, during which time he makes the claim that he can no longer cast magic, without his hair, citing that his hair is the source of his magic powers.

The truth is far deeper than that, though. Quaraunis weak and in frail health, he realis heavily on his stinging, strangling tentacles to survive. His Elf body his badly injured, with a lame leg, and he can barely walk with the Elf's legs. Quaraun moved with his hair, most all of the time, carefully wearing long skirts to hide his feet, hiding the fact that he's actually "walking" on his tentacles and not on his feet. 

But, with his hair/tentacles cut off, he can barely stand, let alone walk. And he hides the fact of his being a JellyFish from most people. Only people he strongly trusts know that he is actually a JellyFish. Thus why the claim that he can not use magic and must go into hiding, after his "hair" is cut.