As others have noted the Djinn are likely not part of the Fae at all, rather they'd be one of the many other denizens of the NN. The Fae are just the closest, and a race unique in that they straddle the boundary and actually natively exist in both worlds, leaving behind corpses etc. Beyond that they varr widely, being depicted as anything from Elemental beings (all wind and sand and fire) to half-animals with human bodies and animal heads (very Egyptian I always thought).
Also, as I recall, one of the only actual qualitative descriptions of the Djinn were that they were VERY specifically in possession of Free Will just like Humans, a specific distinction to the Angels described in the Quran who do not have it.
If they HAD to get crammed into the Fae framework, Id say they'd have to be base Wyldfae and get classified with leaning toward one or the other (or neither) court. Those of Smoke and Fire would be more Summer, while those on Air side might fall to Winter, and the rest would fall out more esoterically based on their "draftable" actions like in SK.