The IRS & SSA systems for issuing numbers are not linked; duplications occur, but they are not frequent. As was done here, the bank should verify both numbers before concluding they represent a duplication. Use of the IRS e-services data query would be the most effective way to verify duplication; i.e. both name/TIN combinations should be reported as "good."
When you file an information return the electronic specifications ask you to indicate the type of number you are using; i.e. ITIN, SSN, EIN, etc. There are no spaces for hyphens in the box for the number so the suggestion that they can tell by the formatting is incorrect.
In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.