No, that’s not correct. There are plenty of multi-engine aircraft without yaw dampers but still have rudder trim, just like some single engine aircraft have them and others don’t. Generally the higher end, high performance singles will have a y/d, but it may or may not be an aftermarket installation.
Yaw dampers are generally coupled to an autopilot, unlike rudder trim, which is manually adjusted by the pilot like any pitch trim. They use accelerometers and rate sensors (like an autopilot would for roll and pitch) to determine motion and then algorithms determine what rudder inputs are needed to keep a turn coordinated and reduce Dutch roll tendencies.