I would never have thought of those real-world parallels, personally. It's a good point.
My take on that was always that it was basically due to simplicity. As the Dominion War story arc developed during seasons 3 through 5, they began to need a sort of catchall term for "us." As the threat from the Dominion expanded out to encompass the fates of more than just the Federation (and of course, since when the real war began, it wasn't just the Federation doing the fighting), the producers needed a term that could just be tossed around representing "all these guys who oppose the Dominion." Technically, Alpha Quadrant does present a problem, since the Romulans, Klingons, and even some Fed core worlds are all in the Beta Quadrant. But what was the alternative? "Our Quadrants"? "Our Half of the Galaxy?" Or, failing that, referencing Alpha AND Beta everytime this subject was brought up? Imagine if every time Martok made one of his grand, sweeping statements about the war, in that distinctive Martok voice, if he said "
The fate of the entire Alpha Quadrant may be at stake! Also the Beta!"
So I think it was just chosen as the best way to get the point across when speaking in these kinds of terms.