ghost Sekai that was "brought back by Akira's deep desire" and the Sekai that "physiologically infested inside Kaori" have technically different yet both valid origins (or as valid as this world allows it).
In other words, on one hand ghost Sekai and Sekaori can still be viewed as two sides of the same coin;
I didn't do a very good job at explaining it, but that pretty much hits the nail on the head. The idea is basically that a piece of the soul, spirit or energy is transfered to the person who receives the organs. I prefer spirit over soul in this case, but words are vague. Anyway this 'dark soul(s 2)' isn't conscious, or mostly isn't conscious, as it's not actually Sekai's soul/consciousness, but a fragment of her. Most likely Just a memory or image of her rather than an actual person.
I'm not a fan of this series, but you could easily liken it to Harry Potter as well. Split off a piece of your soul, it resides in a host, might be able to corrupt or control them to some degree(?) (Sekai = Voldemort? Gonna have to photoshoo that, later...)
There are some problems with this theory, mainly that it's not really distinct or provably different from Sekai actually controlling her in most cases. I think any overt possessions are just Sekai jumping in. Though I do like it, as it gives a way for Sekai's energy/spirit to be "inside Kaori" without Sekai to have to be literally possessing her 24/7, and might explain why and how Sekai can possess Kaori so easily, as she already has a "gateway" into her.
Guess it would be too much to ask for Sekai to be happy for Ami.
Sekai is very mentally unwell. She's at least possessive and likely obsessive, at least post-death. There's also the whole diddling a prepubescent and "falling in love" thing. Which is just fucking gross on top of mentally weird. As Moonflare said (I believe) she's basically here just to fuck with Akira and stay with Ami because she can't let go and to meet whatever sinister ends she has.
Also on a very slightly related note to above, we don't know if Sekai (or all Sekais...) are exactly like her, just passed on now, or if she's been heavily corrupted, or is some kind of evil manifestation of Sekai or something. Most people that die seemingly don't become phantoms that haunt and possess the living. She really is a perplexing, ultra-dead bitch.