What is the Federation's system of government in Kirk's day? And what parallel track did history follow to reach that point?
We know, at any rate, that Soviet Russia must have survived into Kirk's day, even as a sub-state of the Federation. Chekov refers to Leningrad (modern-day St. Petersburg), which would indicate the the Soviet Regime had at least a nominal voice in the Russian government. Obviously, the Federation as a whole is a democracy founded on American ideals, as demonstrated in the episode "The Omega Glory," Kirk can even recite parts of the Constitution! Their history must have diverged from our own after the 1960s, because in the episode "Assignment: Earth," no visible difference can be seen. Obviously, Khan came out of the "Genetic Wars" placed sometime in the 1990s. Science seems to advanced hugely in some areas (interstellar travel!) and yet remained oddly primitive in others (tape-based computers?) Also, the Federation, for all its talk of equal rights of races, seems oddly dominated by humans. Consider the fact that Spock is the only non-Terran serving on the Enterprise!
Given the evidence, in comparison with later series, I would say the Terran Federation is not quite the ultra-pure democracy it is made out to be! Not only do Terrans dominate the corridors of the Enterprise, but every other ship that is seen in TOS has, (at the very least) a human captain.
I would also conclude that history probably diverged around the 1970s, with a scientific emphasis on physics over computerization.
Please reply, either if you agree with me or don't. I am always open to new evidence.