As someone who speaks several languages it is also very apparent that somehow the infrastructure in the brain must be slightly different for different languages. For instance, to pick German again, when "caching" certain parts of sentences for later processing, which does not exist in other languages.
The brain being a muscle, it is only natural that this would translate in more developed cognitive abilities for those language speakers.
I've heard several takes on Asian students being better than others in maths: one of those argued that both the languages themselves, plus the way they are taught to young kids, could, on the long term, make brains better suited to maths. Would be curious to see such studies on that.