Theory: Cape Feathers Produce Wind

Abstract
The Feather item doesn’t give Mario a magical cape that can fly, it merely produces wind that allows him to fly with the cape he already has.

Arguments

 * Argument 1: Mario’s Cape clearly exists independent of the Feather item. In Super Mario World’s manual, Yoshi is the one who gives Mario the cape at the start of the game. In Super Mario World 2: Yoshi’s Island, Baby Mario can be seen wearing the cape when he gets a star. In Mario & Luigi: Superstar Saga + Bowser’s Minions, the cape can be seen hanging on a coat hanger in Mario's house.


 * Argument 2: When the Feather item is used in Super Mario Kart and Mario Kart 8 Deluxe, it does not give the user a cape, but instead produces a gust of wind that lifts their kart up into the air briefly.


 * Argument 3: Wingo has a Feather attached to his turban and has many wind-based abilities. The gem that the Feather is attached to glows and changes colour whenever Wingo uses these abilities. Perhaps he gets his power from the feather itself.