Trey Kings are thought of as demon worshippers, but in reality they serve Therakiel, the Halfway Angel. Therakiel is from Heaven, and isn't connected to the Kings of Edom. The Qliphothic horrors are denizens of the Qliphothic realm, and the ones we see serve the Kings of Edom. They look similar ingame, but it's just coincidence and a little bit of model reusing.
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Thus, all Kings are Qliphothic, but not all Qliphothics are Kings.