My child (age 11) is currently learning vocal, with professional teacher. If we want in the future to do something with this knowledge (join some playing group, maybe proceed with musical education, etc), is it important to start learning some other instrument as well or is the vocal a sufficient on its own?
Strictly speaking, it is not necessary to learn an instrument to become a singer/vocalist. It doesn't help increase vocal range or intonation. Even music theory is not necessary. Music theory helps the singer communicate with other musicians (what key to pick, how many semitones to transpose, etc...) but it won't make him a better performer.
However, learning a instrument (especially one that can play chords, such as the piano or guitar) is one thing that I would recommend to all singers. The benefits are huge (although not directly related to vocal skills):
So definitely, you should encourage him to learn an instrument (piano, keyboard or guitar) but not force him to in case he's not motivated. Make sure you tell his teacher he likes to sing so he can focus on chords. He won't be able to sing and play simultaneously at the beginning, so he'll have to divide his practice time between the two. I also recommend teaching him a couple of simple songs (that he likes) at the beginning (without overwhelming him with theory and technique) so he can experience the joy of being able to play and sing and therefore maintain his excitement and motivation.