As a Mystara fan . . . where in 5E are the shadow elves of Mystara even mentioned, much less equated with the shadar-kai? There are some similarities in skin tone, and both elvish cultures are referred to as "shadow elves" . . . but I missed where they were said to be one and the same.
Shadar-kai are shadow elves because they live on the plane of shadow. Mystra shadow elves are named such because they live underground, in "the shadows". We also have the sith arak from Ravenloft, shadow elves who live in the Shadow Rift . . . these three groups aren't really the same thing, despite the names and skin tones.