Short answer is no.
It is in fact just the opposite. Each bank will have seperate identifier codes not only within a country but also worldwide. SWIFT codes or better said Business Identifier Codes (BIC) have 8 or 11 characters.
BIC-8: BANK BE BB (without a branch code)
BIC-11: BANK BE BB XXX (With a branch code)
The first four characters are unique for the business and chosen by the business when they come aboard among possible 4 digit letter combinations that haven’t been used before. For instance this is BOFA for Bank of America, CITI for Citibank, CITX for CIT Bank, and BKEN for Bank of England.
The idea is that the 4 digit code is the same worldwide (while not always the case due to mergers, takeovers etc)
The next two digits are the ISO code of the country where that business operate. US for US, GB for UK, CA for Canada, FR for France etc