After examining the game code here is what I found:
"The first issue discussed for the new Constitution was the government form. The composition and election of the Senate, how to divide the executive power, between three persons or invest the power into a single president, how to elect the president and how long his term was to be and whether he could stand for reelection.\n\nThe two suggested government forms were 'American Republic', 'American Federation' and 'Constitutional Monarchy'."
This seems incredibly vague, flavor wise, but it discredits the idea of the "American Republic" being the U.S. government under the articles of Confederation, since either unique Government is the result of the Constitutional Convention, in which the Articles are superseded by a new Constitution. That the "Federal Republic" represents the Historical U.S. seems to be supported by this (and the stronger central government it provided helped the U.S. establish itself diplomatically among the European powers, thus the diplomatic bonuses), but what possible historical outcome does the "American Republic" represent? During the Convention the President was originally supposed to remain in office for 7 years, but that was amended to 4 years later; however there was a great deal of things up for debate besides this one point, and often with more than binary options. So what is the "American Republic" supposed to be, government wise, and why does it receive a bonus to STAB and war exhaustion reduction? It does not seem to match the "New Jersey Plan", which was the opposing major outline for the U.S. government; the Virginia Plan being outline that was eventually chosen and thus the basis for historical U.S. government.