The ECB has three functions: a monetary function with both conventional monetary policy tools (interest rates) and unconventional ones (liquidity, asset purchases, etc.). Since banking union was put in place the ECB has also had a micro-prudential function regulating banks and a macroprudential role.
So these three functions give it both an objective of price stability and also objectives of financial stability, but these are clearly distinct and involve different tools. They are also separate in terms of governance, since the EU rules entrusting banking regulation to the ECB make great efforts to ensure that the decision-making processes be kept apart.