Are there bands specifically dedicated to anime?
Yes.

Though they are not actually an actual and active band in Japan but Houkagou Tea Time (HTT) (composing of the K-On!'s characters' voice actresses) were created for the anime's purpose itself. Toyosaki Aki (Hirasawa Yui), Kotobuki Minako (Kotobuki Tsumugi), Hikasa Yoko (Akiyama Mio), Satomi Sato (Tainaka Ritsu) and Taketatsu Ayana (Nakano Azusa) actually performed (sung while playing their character's respective instruments) some of K-On!'s OST live during two of their live concerts, Let's Go and Come With Me. The voice actresses were taught how to play their characters' instruments for the live concert so I think HTT was specifically created in dedication for the anime K-On!.