I wouldn't be handing my DD's birth certificate over but would go with them to the bank so the bank can have a copy if its needed.
Maybe they want to put it in your DD's name for a tax break or something?
I'd be more concerned about identity fraud than anything else, how do you know what they will use their copy for in the future? or whose hands it will pass in to?
A BC is not something to hand over to anyone.
Maybe its cause my DH works in fraud for a financial institution that makes me skeptical but I certainly would not be handing a Birth Certificate to anyone.

