From the wiktionary discussion about drama queen:
Seeing as the term has been stolen from Gay urban culture, It is never a woman. Queen (derived from 'Drag Queen' or female impersonator) or simply an effeminate rather than the more common, masculine type of gay man.
The origin of the term "queen" is not about a woman. Rather, a man. It should always remain in the form "drama queen" regardless of gender, and can be applied to either a man or a woman.
In colloquial usage, it's almost never converted to 'king'.
Perhaps a small number of people, even non-native speakers, will be confused, and they will generate alternative forms which are then found on some webpages.
In an ngram chart, "drama queen" is 30 times more common than "drama king". The idea or the concept of "drama queen" can apply to men and women. Logically, if "drama king" were the correct terminology, if "drama king" was the male version, then the ngram chart would show a closer level of parity. But it doesn't. That is because the "king" alternative is a joke, a "play on words", or a mistake, rather than the ordinary phrase.