The Bank of England, public finance, and naval power

Was the Bank of England created to build the Royal Navy?

No. It was created to organise government finances, which in turn helped the state create and support the Royal Navy.

The Bank of England was created to put government finances in order rather than solely to build the Royal Navy. A stronger fiscal system did assist the creation and support of the fleet, but that naval effect followed from the wider financial function. The distinction matters because an institution can strengthen sea power without existing solely as a naval organisation.