Removed speculation

Kingdom Hearts:

*The music references Kairi's debut in ''Kingdom Hearts''.
No it doesn't, all of the music debuted in the original Kingdom Hearts, so this isn't an outright reference.
*The first battle condition references how Sora must search for and protect Kairi in ''Kingdom Hearts''.
You could also say that this is a reference to how Kairi and Sora are friends, or how it's a reversal of Kairi jumping in to save Sora in Hollow Bastion, or how Sora and Kairi team up in Kingdom Hearts III Re:Mind. The point is that you can't solidly claim to know what is fully being referenced.
*The Health Recovery rule references the Cure spell in the ''Kingdom Hearts'' series.
This is flat speculation, just like the above point, you can't claim to know what's being referenced with something like this. It could be any number of things, like multiple health bars.
*The music references how Terra visited Radiant Garden prior to it becoming Hollow Bastion in ''Birth by Sleep''.
Terra visits Radiant Garden once in that game, and while, yes, there's an important plot element there to be seen, it's for like 5 seconds and the story moves on from there. I don't know why they chose the theme that they did, but it's certainly not a reference to that.
*The music references Axel's status as a boss in ''Kingdom Hearts II''.
*The music references the battle between Ventus and Vanitas in ''Birth by Sleep''.
Again, music limitations. It just happens to be a boss theme.

Sonic:

*The stage references Tails' appearance in ''Sonic Lost World''.
They just happened to pick one of the only two Sonic stages for this fight. You could have picked Green Hill Zone, despite Tails' debut in Sonic 2, and still had it somewhat work because he's the character you save after going through the level in Sonic Generations. This isn't a reference.
*In World of Light, Tails' spirit is located on Galeem's side on The Final Battle map while Dr. Eggman is featured across from him on Dharkon's side, referencing their shared genius and rivalry in ''Sonic Adventure 2'' and ''Sonic Lost World''.
Like before, you could easily say this is a reference to a number of things. Tails is a hero, Eggman is evil; they both go through the same stages and play similarly in Sonic Adventure 2; they're both close to Sonic and can fly around.
*The stage references Super Sonic's/Amy's appearance in ''Sonic Lost World''.
See the Tails point.
*In World of Light, Rouge's spirit is located on Dharkon's side while Yarn Poochy is featured across from her on Galeem's side, referencing how both characters are treasure hunters, though Rouge hunts treasures for selfish reasons.
Two characters from different franchises and companies happen to share one single character trait? What a surprise!
*The music references Blaze's appearance in the Nintendo DS version of ''Sonic Colors'', which plays similarly to Blaze's debut in ''Sonic Rush''.
*The music references Silver's appearance as a boss in ''Sonic Generations''.
Both of these are a big stretch, especially since the Wii version of Colors lacked Blaze (and the rest of the familiar Sonic characters, barring the big three) entirely. (Though, I'm surprised they didn't give Silver "His World", but I digress.)
*The music references Infinite's appearance in ''Sonic Forces''.
...you mean, the only game that Infinite has appeared in?

Banjo-Kazooie:

*The stage and music represent Tooty's appearance in the title sequence of ''Banjo-Kazooie''.
*The music references the Jiggy's debut in ''Banjo-Kazooie''.
*The music references the Buzzbomb's appearance in ''Banjo-Kazooie''.
Yet again, these are because of music limitations, and even more so given that there's only four Banjo-Kazooie games, half of which have songs pulled from them in Smash.
*The battle conditions reference how Tooty is rescued from Gruntilda's imprisonment at the end of ''Banjo-Kazooie''.
...how, exactly? Is it just because you're fighting the puppet fighter possessed by the Tooty spirit? Because that's a stretch if I've ever heard one.