It’s not exactly the same. Authentication with SSN is simple - SSN + date of birth/name/mother’s maiden name. All of these are immutable so you’d better hope that any firm that collects this information is careful with it. With Aadhaar, authentication is marginally better - either with biometrics (fingerprint or retinal scan) or with an SMS OTP. Identity fraud is still possible but it isn’t scalable, thankfully.

There is still a concern though. If every service that you consume uses Aadhaar as a primary key, it becomes trivial to track the movements and activities of every citizen. I’m certain that for a fee to a shady person you’d get a dossier of places a person has been, where they’ve stayed, who they’ve spoken to, what they’ve spent money on. That’s not theoretical - you can get this info on a Chinese citizen today for less than $100 if you know whom to talk to.