no, you manually need to put them on their specific costume if you want to do that

Every song will default to the character's default, but there is an intended costume for each one. You will have to assign it manually, but it should stay set for the song once you've set it as long as you have them set to Song Specific. (Click right Thumbstick when choosing a song) When browsing costumes, you can press L2 and R2 together to instantly jump to the recommended costume for that song.