@Selaphiel
There’s also Iris Libre, for non-Vichy France.
@neutralgrey
Short answer: you’re primarily playing from the perspective of the Eagles fighting the Sakura, who are aligned with/under the control of the Sirens. Iron Blood is also aligned with the Sirens, but since they’re over in the Atlantic, you only see them during events. Even though the factions are the bad guys, individual IJN or KMS ships may still be good guys, either in the story or just because you have them in your dock.
Long answer (as far as I know, the story isn’t 100% clear): The ultimate villains are these guys, the Sirens:
They appeared sometime in the distant future and pretty much steamrollered humanity. Apparently that was too boring, so they went back in time and started messing with history, most notably by introducing shipgirl technology to humanity. After that happened, the major navies of the world formed the “Azur Lane” alliance for mutual defense against the Sirens. That went along fine until the Sakura (nominally led by Nagato, but effectively controlled by the fox sisters, Akagi and Kaga) and Iron Blood (led by Bismarck) split off, formed “Crimson Axis,” and allied with the Sirens. Those two factions always had a different idea of how humanity should be using/developing shipgirl tech, and eventually their overall mindset shifted to believing that an alliance with the Sirens was the best way forward. (You can see the factions’ philosophical differences approached in how the girls look: the Allies preferred to keep their technology as human-designed as possible, so they’re mostly normal humans with normal looking rigging, the Sakura mixed magic with their shipgirl development, so they’re animal or oni girls with normal or magical looking rigging, and Iron Blood tried to go as close as possible to the Sirens’ technology, so they have bio-mechanical monsters for rigging.)
Not all of the individual shipgirls followed their faction (like Akashi or Mikasa), and as time goes on, some of them defect back and start fighting the Sirens again (like the crane sisters, Zuikaku and Shokaku). The Axis ships in your dock either fall into one of those categories or are clones/reborn versions of enemy ships. I’m not exactly sure how that last one works, though.