Cloud is in Spirits mode, though.
Yeah, he sure is in the second portion. Take it from both him and me, since I actually just finished Adventure mode last night. Basically, the game tricks you by...
Spoiler, click to toggle visibilty
... throwing you into a second portion after defeating Galeem, which introduces a brand new character, Dharkon. He, contrasting Galeem, controls all the darkness in the world and you must travel to the "Dark Realm" to defeat him.
The portion where Cloud pops up is in ...
Spoiler, click to toggle visibilty
... the second portion, as aforementioned. One of the mini-realms you need to visit in order to fight Dharkon is a realm in the theme of the Zelda series, where you have to collect the three triforce pieces. By "awakening" the three unlockable characters in there, they will each hold a triforce piece needed to both unlock Ganondorf as a character and fight Ganon, the boss equivalent. Cloud is represented as Adult Link (Hero of Time, holder of the Triforce of Courage) since all three Links have been unlocked at this point.
So yeah, he's in.
(a thing was here)