The door bell is powered from a low voltage AC source, the EZIOxx family of devices Inputs are looking for DC or a dry contact switch. You can use a relay with an AC coil that matches the voltage of the door bell. The relay coil is connected in parallel with the door bell such that when the bell is energized, the relay is energized. The NO or NC relay contacts are connected to an Input, just like the magnetic switch.
I used a low voltage AC relay powered by a door bell transformer when using an EZIO8T which uses Triacs to drive the outputs. Smarthome carries an AC relay, part # 7195A. They are on the expensive side because the relay contacts are rated for 10 amp AC. I’m sure there must be less expensive relays from an online source. Smarthome was just convenient at the time.
You could also create an AC to DC circuit, powered from the door bell, such that you have a DC voltage to connect to one of the opto-isolated Inputs when the door bell is energized. I think the relay option is the easier of the two approaches.