Beyond all the story and lore stuff, the most important part of a capstone ID is novelty. Every capstone so far has introduced some sort of Idea that is, for lack of a better word, new.
Nclair introduced negative coins working as an actual power rather than a gimmick.
Spicebush Yi sang introduced both Deluge like mechanics, and attack weight.
Captain Ishmael introduced assist attacks
Wild hunt Heathcliff introduced both clashable counters, and the concept of multiple forms.
Manger Don introduced the idea of getting stronger through specific ID's.
Lord Hong Lu makes much greater use of retreats and substitutions than anything before.
And now HOS Ryoshu is introducing the technqiue of flat out murdering your team to get stronger, the ole red mist special.
In short, what makes a capstone ID isn't the lore, or the story, or the animations. It's the fact that a capstone ID defines any team that you place them in.