I have not read the books so my knowledge is limited, but they seem like a financial institute with great power, even power over the Iron Throne. Were there any organizations in the middle ages with similar reach/power?
For accuracy's sake, the organization in the show Game of Thrones and the books series A Song of Ice and Fire, is known as The Iron Bank of Braavos. This Bank did hold considerable power in the terms of wealth, and was also located on the powerful island "Kingdom" of Braavos.
This of course doesn't answer the question but does add more background information for anyone with the right knowledge to better "answer" the question.
I haven't read the books, but I know that some Jews in Middle Ages Poland had considerable influence on both the nobility and the king. They were originally brought in to help settle and develop the region after the Crusades (the Middle Eastern ones, not the Northern, though that may also have played a factor), and curried favour with Polish royalty because they would readily lend money. They soon were hired by nobles to run estates on their behalf (the nobles, owning vast swaths of land, were unable to rule directly), creating animosity towards the peasants and clergy. They also faced opposition from Christian merchants, who tried to force Jews out; they were able to bar them from many Polish cities, but the Jews managed to utilise their connections with the nobility to stay competitive, often underselling the Christians. This declined after the Thirty Years' War of 1618-48, the Khmelnitsky/Chmelnitski Uprising of 1648, and various wars with Russia soon after; the Jews lost a lot of money and influence due to massacres and the destruction of parts of Poland.
Sources:
Davies, Norman. God’s Playground: A History of Poland, Volume 1: The Origins to 1795. Revised edition. New York City, New York: Columbia University Press, 2005.
Johnson, Paul. A History of the Jews. New York City, New York: Harper & Row, 1987.
Margolis, Max L. and Alexander Marx. A History of the Jewish People. Philadelphia, Pennsylvania: The Jewish Publication Society, 1947.
Stone, Daniel. The Polish-Lithuanian State, 1386 – 1795. Seattle: University of Washington Press, 2001.
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