VReg with MIC5156, why a so high drop-out voltage?
The MIC5156 does not include an internal charge pump:
Block diagram:

The datasheet also has this to say:

Version Differences
The MIC5156 requires an external voltage for MOSFET gate drive and is available in 3.3V fixed output, 5V fixed output, or adjustable output versions. With 8-pins, the MIC5156 is the smallest of the Super LDO Regulator Controllers.

This gate supply is derived from the Vp pin (note the +12V input applied in the application schematic above), which must high enough to be able to turn on the FET properly.
As you are supplying this pin with the same voltage as all the other pins you will not get low drop-out regulation across the MOSFET until Vp exceeds the necessary Vgs to turn the FET on.
In your case, this is apparently 1.8V (but it could be as high as 4V or more according to the datasheet).
Solution:

Drive Vp with a higher voltage than supplied to the drain of the FET, with enough headroom to turn the FET on properly.

[Update]: Hat tip to FakeMoustache for another solution:
Use the MIC5157 and add the necessary charge pump capacitors. (see datasheet).