If Aadhaar has to be freely used for proving identity and it is safe to do so, then why has UIDAI advised people not to put up their Aadhaar number in Social Media or public domain?

You use PAN card, debit card, credit card, bank cheques wherever required. But do you put these details openly on internet and social media such as Facebook, Twitter, etc.? Obviously no! You do not put such personal details unnecessarily in public domain so that there is no unwarranted invasion attempt on your privacy. The same logic needs to be applied in case of uses of Aadhaar.