Yes and no? In a way of course everyone is reincarnated, that's just the way of life. But i know you're asking bc of the earlier question, so the answer is that most of them are indeed reincarnated, but some like Reef, Yasumi, Ammianus and Bruno aren't. Violet is also technically not reincarnated, because she is a digital copy of a brain that was put into a body, so she was revived, similar but not the same.
Other's have partial reincarnations like Gemma and Shellie, where they aren't aware of it outside of some deju vus and 'corrupted' memories like a computer.

On how it works... i never really put any thought into it, but whether someone remembers everything from their previous life or just some parts is totally random. But with enought tries they can remember more. As for the way they're reincarnated, it's mostly because of the Alterna Liquid crystals. Ya know the thing in the Alterna logs that made marine life evolve so much in 12 thousand years? Yeah they canonically contain human desires and dreams, and in my fanfic's canon they serve as a connection between the different universe versions of everybody, so only people that contain a lot of it in their dna can actually reincarnated. For example: Jack originally wasn't reincarnated as a Salmonid, only after contact with the crystals when his family were near the splatlands did he reincarnate.
Another requisite for reincarnation is that the person has to obviously be dead or in a prolonged coma.
I hope this is a good explanation :D