Why is there no black dragon in Mahjong?
The simplest answer is that the "Dragon" tiles are more of a triplet, and not a pair.

This is pretty clear when using one of their Chinese names 三元牌 or Three Fundamental Tiles,
the name "Dragons" was added by Joseph Babcock when introducing the game to America (he also mistakenly named the 索子 suit bamboo).
When translating the game Babcock should probably have stuck with the most common name, Arrow tiles.

The White Dragon was the first of the set added to the game, originally using replacement tiles (fully blank) as it's representative; it was only once the Red Dragon, and then later Green Dragon tiles (~20 years) had been added that the White received additional markings to differentiate it from other replacement tiles.