Not quite. In this case, “to break out” is to become popular. In your example, either a song or the band can break out. I assume in your sentence there is just one song: the song you listened to is the same song that went viral on TikTok. If that’s the case, you might mean: “Today, I listened to the song by a local band that broke out on TikTok.” (This can mean either the song broke out or the band broke out.) You can also say something like, “Today, I listened to that song that broke out on TikTok. It’s by a local band.”