Does a Shanghai social security card have to be from a Shanghai bank?

Not required.

The new version of the social security card can be placed under the service bank:

The new version of the social security card service bank through public bidding to determine the way, including: Bank of Shanghai (95594), Shanghai Pudong Development Bank (95528), Shanghai Rural Commercial Bank (962999), Industrial and Commercial Bank of China, Shanghai Branch (95588), Agricultural Bank of China, Shanghai Branch (95599), China Construction Bank, Shanghai Branch (95533), China Postal Savings Bank, Shanghai Branch (95580), China Bank of Communications, Shanghai Branch (95559), Bank of China, Shanghai Branch (95566), China Merchants Bank, Shanghai Branch (95555), China Everbright Bank, Shanghai Branch (95595) and other 11 Banks.

Applicants can voluntarily choose a banking service among the 11 banks when the new version of social security card is issued.