Sorry to say I disagree. Given it seems to be some kinda homework or exam test (so we don't know how strict are the requirements), I2C is a synchronous serial interface, because it has master and slave roles, and a clock (SCL). Even Wikipedia confirms it is a "synchronous, multi-master/multi-slave (controller/target), single-ended, serial communication".
So IMHO implementing I2C communincation between two Arduinos looks like it complies with the original requirement.