Under the requirements of the Patriot Act, all banks must run an identity check on any new customer. They use "customer identification programs" (CIP) that compare your name, address, date of birth, and other facts to the information contained on your credit report. Your credit score is not damaged by this step.
The CIP regulations require institutions to implement reasonable procedures
Verifying the identity of any person seeking to open an account, to the extent reasonable and practicable
Maintaining records of the information used to verify the person’s identity, including name, address, and other identifying information.
Banks also review ChexSystems and sometimes credit report data during the process of determining whether a customer qualifies to open a new account. Under FCRA regulations, the bank has to have a permissible purpose such as "intend[ing] to use the information in connection with a credit transaction involving the consumer" in order to pull a credit report and must obtain the consumer’s consent first. A hard inquiry like this could cause a small credit score ding.