Aside from nitpicking (which don't matter at all like replace passport with Aadhar, would you believe it then?), the story have nothing unusual - just those people had this person in mind due to media's 24/7 propaganda.
Ok, I'll list out the suspicious points so that it will be easier to analyze:
1) Random person comes and asks whether he is a Muslim (In an
Upscale Restaurant, the crowd is more mature, the hotel staff won't want any commotion, etc)
2) He conveniently forgot his Aadhar card (You asked whether I would believe if he had Aadhar card, but over here Aadhar card is not the point of contention but only one of the many loopholes in this story). The Aadhar card is a more secure identity and in fact, one of the most important document to identify any person in India.
3) He mentioned that he had stayed in Jaipur for over a year and yet had just '
forgotten' his Aadhar card when he left his native place for Rajasthan. Any person in India will be careful to take his Aadhar card when he travels around even within his own state. This careless guy forgot his Aadhar card and thought it's ok to live without Aadhar card in Rajasthan.
4) In many hotels and apartments, you are required to show your Aadhar card as proof to live over there. How didn't he have his own Aadhar card? A person in his 20s/30s is supposed to have his Aadhar card right? Did he just lose it and was careless enough to not a create a new one?
5) In cities especially, people are not too concerned about your identity. People tend to keep to themselves.
6) All Hindus know mantras. Even if he is an atheist/agnostic, he can still easily recall the mantras that he used to recite in home/festivals when he was a kid.
7) It's mostly the Brahmins who wear the janeu. The people really asked him about janeu? So can we classify this incident of a person being beaten for not being a Brahmin instead of for not being a Hindu?
8) Apparently, they checked for circumcision (which was caught on camera with the hotel staff idling around lol). Ok, so they must have found he is not circumcised. So why continue beating him?
9) Heavily accented Hindi? There are plenty of people who don't speak Hindi with correct accent from all over India. Going by his name, he sounds a Bengali. A Bengali speaking Hindi could be very different from how a Kashmiri speaks Hindi.
10) How do documents from National Crime Records Bureau website translate to people assuming he is a terrorist? What's the correlation?
I don't think we even need to go to the police station part from here.