Keeping in mind that a plane can fly with different engine models (e.g. for test purposes), has an error ever happened during normal operations that led to the wrong engine being mounted? Is some kind of calibration or check necessary before flight to ensure that the engines are the correct ones?
January 2016: a "wrong" or non-functioning engine was mounted onto a Qantas 747. QF63 departed from Sydney to Johannesburg with a 5th engine.
The 747 is capable of carrying these kind of loads through anchor points on each of their wings.
Disclaimer: this wasn't by mistake, it was intended to get the "5th" engine to Johannesburg ASAP, to be fitted onto another aircraft.
As for the possibility of mounting an incorrect engine on an airplane by accident, it's very unlikely (impossible really). Jet engines are designed exclusively with the aircraft in mind. Further, jet engines are expensive you would make certain you order the correct engine for the corresponding aircraft. Aviation mechanics also receive sufficient training and the an engine replacement process isn't a one man job. It circulates through a group of individuals prior to actually making a purchase or deciding to fit a new engine.
Besides, even in the case where some unqualified individuals were to attempt mounting an incorrect engine, it wouldn't work. First, they would easily be able to distinguish the existing engines from the replacement. Second, the mounting the engine in the place wouldn't work, the mount points would be of a different shape and would consist of bolts and screws in different locations.