Translation:His hobby is listening to music.
Not a native speaker here, but they seem equivalent to me.
I reported and asked that "to listen to" should be approved as well.
A native English speaker would never say it that way, at least in my area. Then again, I don't think a native speaker would ever call listening to music a hobby, either, so I think it should be close enough for Duolingo to mark it correct.
Yes it is grammatically correct, but in this case it is probably better to say "listening" because it is his hobby and therefore it is likely to be an ongoing or habitual practice, and so the continuous form of the infinitive is used. Using "listening" makes it clear that it is referring to something that he is already doing. However if you were talking about something he wanted to do in the future you would be more likely to use the "to" form of the infinitive e.g. "his ambition is to become a ...." because he is not doing it yet.