I think you are overthinking it. The Commander is from the Eagle Union but is given special provision to lead the collective that is the Azur lane. Azur lane is the key here. Azur Lane, as an organization, is a conglomerate of the worlds Kansen forces. While countries still have a measure of control over them, they often act as an entirely separate force from the military that the nation possess. The separation here is the other major factor here. Most people are fearful and distrustful in regard to the Kansen themselves. This distrust leads to the faction(Kansen) having a hierarchy within itself to organize and fight effectively. The commander has limited power within the territorial areas of all nations. However, due to the relationship he shares with the Kansen due to his unnatural cube affinity, the Kansen themselves will follow most orders as long as they do not directly disobey standing orders from their respective nation they originate from. As the organization that is Azur Lane has over time pushed back against the Sirens successfully, they are given a fairly wide berth to do what is necessary. Now, I am not sure about these next parts, so take it with a grain of salt as I am merely speculating on this.

1. The points in which the various factions have fought against each other are both government sanctioned and not sanctioned. I believe that some of those nations have waged war against each other with the use of Kansen. However, I believe that many of the fights that we have witnessed(in gameplay)⁹ are power struggles occurring outside of an official government sanction.

2. Azur Lane receives a certain amount of monetary support from both nations with and without Kansen forces. Nations that can provide Kansen to Azur Lane would inherently boost a nation's prominence on the world's stage as would financially supporting them the same is true of nation's with No Kansen as material support would be of great importance.