Imo, its due to damping factor. The additional power at higher gain can make some headphones sound more effortless (best word I can think of).
For the iDSD micro, the different power modes do not change either output impedance / damping factor (damping factor is load impedance divided by output impedance). Neither does the current limit of the circuit change.
The output impedance / damping factor and current limit change only if iEMatch is engaged.
The lower power is simply due to reducing the power supply voltage and the circuit gain, thus making the amplifier clip earlier but also consuming much less idle power.
The largest objective change we observe is that the higher gain / power modes have marginally higher distortion levels into low impedance headphones (< 50 Ohm) and absolute background noise is raised in line with gain.
So, objectively the higher gain/power modes produce more background noise, a smidgen more distortion and the amplifier clips at higher output voltage levels.
What this means is that the higher gain/power are matched to higher impedance, lower sensitivity headphones. In this case the distortion remains low and the noise at the actual headphone accoustic output also remains low and the added output voltage compensates the lower sensitivity.
We would not wish to venture speculations as to why some iDSD micro users perceive large sonic differences with power mode.
However our own listening panels (using multiple iDSD micro accurately level matched) did not pick up on these differences and which show no evidence in measured performance.
It is essential to make sure to adjust the volume setting using a test signal and a suitable signal meter to compensate gain changes, rather than attempting to level match "by ear".
As little as an 0.5dB higher level (and 0.5dB are near impossible to set accurately and repeatedly by ear) will give a sound that is subjectively perceived as more dynamic and possibly with more bass, but is not perceived as louder.
It takes around 3dB level difference before the louder source is identified as louder using music, so it is relatively easy to be fooled by small but significant level differences into "hearing things" that in reality do not exist.