Is "Kiyosumi High School Mahjong Club" from Saki a real-life organization?
It's not too uncommon for anime production committees to be named after something in the show. In Saki's case, it is named after the high school where the show takes place. This does not mean that a real life group from the school (if it even is a real school or club at said school) has anything to do with the creation of the show.

Essentially, the production committee is the rightholder to the series rather than a person or group who had a hand in creating the series (Source: a JETRO pdf about the anime industry, specifically page 4). It can consist of all the companies involved in an anime - the animation studio, TV channel, merchandise creators - who want to see the show succeed because it will benefit them (Source: an ANN feature about the anime industry).

Rather than list all of these companies, it is easier to give them one name, and arguably more entertaining to the viewers if it references something in the series. When the production committee isn't named after something in the series, it is usually just called "Series Name Production Committee" - in your case, it would've been called "Saki Production Committee".

Here are examples from a few other shows:


Seitokai Yakuindomo: Ousai Academy Student Council Room(the show is about Ousai Academy's student council)


Steins;Gate: Future Gadget Lab(the show is about a some eccentrics and scientists who are part of a group called Future Gadget Lab)


Tamako Market: Usagiyama Shopping District(the show is about the people working and living in Usagiyama Shopping District)