If you're looking for anime similar to IDOLiSH7: Vibrato, you might like these titles.
The members of Jupiter — Amagase Touma, Mitarai Shouta, and Ijuuin Hokuto — are aiming to be the best in the idol industry. As their concerts start selling out and their popularity increase, they experience issues.
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Both of these are spin-off prequels that follow the creation of a three-man idol team, and their rise to fame. Both are great introductions to their own franchise. I'm confident that if you enjoyed one, you'd enjoy the other.
After appearing in the Prism King Cup representing Edel Rose, and getting a glimpse of what it was like to a be a future Prism Star, Ichijo Shin entered Kakyoin Academy’s High School division in spring and became a first-year high school student. Koji, Hiro, and Kazuki from Over The Rainbow left Edel Rose and established a new agency. Meanwhile, after having the title of Prism King stolen by Hiro, the leader of Schwarz Rose, Norizuki Jin, proposes PRISM.1, which would decide a new king other than Prism King. He then announces that Schwarz Rose and Edel Rose will be competing with each other.