Kirby's yellow costume is Keeby. It is also the same as when he is Beam Kirby, sans the hat.
Kirby's red costume is the same as when he is Fire Kirby, sans the hat.
Kirby's blue costume is the same as when he is Ice Kirby, sans the hat.
Kirby's green costume is likely an homage to his days on the green Game Boy screen. It also resembles Plasma Kirby, sans the hat.
Kirby's white costume is a reference to his monochrome appearance from the original Kirby's Dream Land for Game Boy. Coincidentally, it is also similar to the colors of Kirby's depiction on the North American box cover of Kirby's Dream Land.
Mario's yellow costume is based on Wario, the same as his previous yellow costume.
Mario's blue costume is a red/blue reversal of his traditional colors. This is seen as a partial reference to Mario's varying appearances in games prior to Nintendo's officially establishment of Mario's outfit of consisting of a red shirt, blue overalls and red hat. In various earlier games, Mario occasionally also appeared with sometimes a blue hat, red shirt with blue overalls or a red hat, blue shirt with red overalls. Interestingly however, Mario never appeared in any game with the particular combination of a blue hat, blue shirt, with red overalls as it appears here.
Mario's brown and white costume is a reference to Foreman Spike from Wrecking Crew.
Mario's green costume is a color inverse of his traditional colors. Contrary to popular belief it is not a reference to Luigi, however it does coincidentally resemble Luigi's colors from the arcade version of Mario Bros.
Peach's yellow costume changes nearly her entire appearance to that of Daisy, including her hair color, hair style, earrings, dress style, gloves, crown and even skin color.
Her blue & green costumes imitate Mario & Luigi's caddies in NES Open Tournament Golf.
Peach's white costume represents how she appeared in her first appearance in Super Mario Bros. Although Peach always has naked legs in her appearance in many games, she's wearing white tights with this costume, which doesn't happen with her other costumes.