So, I just watched every cutscene in KHIII again, and the thing that really struck me this time is how consistently the game focuses on self-sacrifice and death.
Apart from KHI and BbS, KH's choice of Disney worlds seems to be rather haphazard -- the real story happens in the original worlds, and you get involved in random-seeming Disney plots in between. But, while there's definitely still a lot of filler in KHIII's Disney worlds, there's always a solid thematic hook:
1) Olympus - Herc's big self-sacrificial moment may have already happened in KHII, but it's the reason why Sora goes to Olympus in the first place, and is the reason Herc is able to give Sora the most important hint towards using the power of waking ("with all your heart"). Xigbar also hints at Sora's ultimate fate, questioning the value of risking one's life for others when that will mean others will have to risk their lives for you.
2) Toy Box - The self-sacrifice theme isn't in focus here, but the idea of toys having hearts becomes important later anyway, given the importance of Replicas.
3) Corona - Flynn sacrifices his only known opportunity for healing to ensure Rapunzel won'd sacrifice herself to heal him, and Rapunzel is able to use her healing powers because she is willing to give up everything to save him. This is directly referenced when Kairi guides Sora back after he saves everyone else's hearts (along with a similar scene from Frozen), which makes me wonder if this might be the nature of the powers granted to the New Seven Hearts.
4) Monstropolis - Like Toy Box, the self-sacrifice theme isn't in focus, but the concept of using negative emotions as power is perfect for explaining the presence of Vanitas after his loss in BbS.
5) Arendelle - The key hint in Arendelle's plot -- "only love can thaw a frozen heart" -- refers to Anna's self-sacrificial love for Elsa, not romantic love (as everyone expected). This provides a surprisingly effective parallel to Ven's awakening, which is triggered by Aqua's self-sacrificial willingness to take a fire spell for him.
6) 100 Acre Wood - the key concept here seems to be the idea that what's lost can always be found again, somehow. I initially thought that there had to be more to what happened to Pooh's connection to Sora than what they showed, but on a second viewing, it seems more like it's meant to hint at Sora diving in to save his friends' hearts in the ending.
7) The Caribbean - Death is everywhere in PotC3, and KHIII takes advantage of that. Sora has to help Jack escape the afterlife, then ends up stuck outside the movie's plot until the final kill-or-be-killed showdown with Davy Jones, where Jones flat-out murders Will Turner in front of Sora and triggers Sora's inability to cope with loss for the first time in the game.
8) San Fransokyo - It's impossible to avoid Tadashi's death if you're going to have any meaningful characterization for the Big Hero 6, and Repliku 2 takes the loss theme up a notch by forcing Sora and Hiro to disable the first Baymax and destroy his chip.
Then, of course, once the original worlds start appearing, you've got stuff like the Final World and the Lich (whose attempt to take Sora's lost friends to "the abyss" makes me wonder if that might be KH's equivalent of Hell), MX literally turning Kairi into a human sacrifice to summon Kingdom Hearts, and Sora's choice to use the power of waking improperly knowing that it might cost him everything. Sora telling Xemnas that "pain is just being human" seems related, too.
All in all... KHIII seems rather thematically similar to FFXV...? The primary concerns, at least, also appear to be love and sacrifice.