As your bank information is not shared by the bank with anyone else, no one can get to know your bank account by knowing your Aadhaar number.
Also, UIDAI or any entity for that matter will not have any information about your bank account.
For example, you give your mobile number at different places and to different authorities like banks, passport authority, income tax department etc. Will the telecom company have access to your bank details, income tax returns, etc.? Obviously not!
Similarly, when you provide Aadhaar number to different service providers, your details remain with the respective service providers and no single entity including the government or UIDAI will have access to your personal information spread across different service providers.
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