The GTVA is, above all else, an alliance, so its in fact two seperate governments who work together to handle threats to both parties aka the Shivans. The Security Council is little more than the group who decides when and why the articles of the alliance are necessary, then sells the idea to the two independent governments.
The Terrans have the old core of the GTA to run off of, which was likely more of a confederation than a federation. Of course, after losing Sol, they probably leaned much further into federation status, sort of like the US, though more likely a parlimentary style government, except with an elected President of some type.
The Vasudans have an empire, ruled by an Emperor. They had a large scale elected government, until it was desolved after the Great War (probably over the parliment wanting to reignite the war with the GTA, something that GTA officials would have also tried to pursue if Sol hadn't been cut off). Thus, beyond very local officals, people are placed in positions of power by order of the Emperor.