No. As your bank information is not shared by the bank with anyone else, no one can have information about your bank account just by knowing your Aadhaar number. Also, UIDAI or any entity for that matter would not have any information about your bank account. For example, you give your mobile number at various places and to various authorities such as bank, passport authorities, income tax departments, etc. Would the telecom company have access to your bank information, income tax returns, etc.? Obviously no! Similarly, when you provide Aadhaar number to various service providers, your detail remains with the respective service providers and no single entity including the Government or UIDAI will have access to your personal information spread across various service providers.
The UIDAI asks the resident to confirm the addresses (from the multiple address proof) on which he/ she wants his/ her Aadhaar letter to be delivered. Based on resident’s choice, UIDAI captures the detail as per supporting documents.