What can cause the output voltage of a buck converter to drop (apart from high current)?
Mystery solved.

Problem was caused by the STBY pin of the converter.

This STBY pin puts the converter in standby mode. 1 = standby and 0 = on. When the MCU is powered off, the floating pin is interpreted as 0 and everything works fine (V=1V8). 

I am not sure what is the factory default configuration of the MCU, but it seems that when the it is powered on, the STBY pin is interpreted as 1 from the converter, essentially turning the converter off. This leads to the converter going on and off continuously.

I verified this theory by grounding the STBY pin manually, with the MCU connected, and the voltage went up to 1V8 (with current at 2-3mA).