It is an assumption that naruto's chakra cloak is what gave sasuke's susanoo legs. But it is an assumption asked off of manga facts. It is a fact that naruto's chakra cloak powers up the abilities of the people he gives it to. Example being kakashi kamui being 3 times stronger or hinatas air palm being 3 times stronger.
It is only plausible to assume that sasuke has gotten the imperfect susanoo due to naruto's chakra cloak. We do not know this to be 100% true this is why but it is highly likely it is.