It's actually quite clear. I have been there twice, and the first time, I obtained permission to stay from the Patriarch rather than the Pilgrim Office in Ouranoupolis, and thanks to that letter, I was able to stay for the whole of Lent rather than just the two nights the Sacred Community of Mount Athos usually only permits.
Is the Ecumenical Patriarch a hybrid Head of State if anyone can directly ask Him for Permission to stay on the Holy Mountain and circumvent two nights maximum stay if one asks the Mount Athos Administrators? If the answer is yes, are the monastics justified in saying that the Ecumenical Patriarch has declared Himself Pope of a sovereign area much larger than the Vatican?
Why don't you ask a less obtuse question, like "why can he do that?" rather than the seemingly-inflammatory ones above?
Mt. Athos is a unique place, which both contains
monasteries, and is (for all intents and purposes) also a monastery itself. It is a place governed for the monks, by the monks. The EP's authority at the essential level
is that of a Bishop over his monasteries. The jurisdiction of the EP over the Holy Mountain is probably
due to the close imperial ties to the region during its great expansion (otherwise, it should probably be under the Archbishop of Thessaloniki or someone like that).
However, additional responsibility/authority is granted by the State of Greece, which retains highest civil jurisdiction (hence, why it is called "Semi-Autonomous" in a legal sense) through its constitution, by explicitly maintaining that only those in communion with the EP have the legal right
to occupy the Holy Mountain. This is essentially where the crux of the current dispute lies. See, any monk can be asked to leave a monastery by the Bishop under whose spiritual direction the monastery is under. However, only on Mt. Athos does said eviction also carry the weight of law and the enforcement of the state. The enforcement by the state is what is being decried as inhumane - because the state does not feel compelled to abide by the same standards of charity that the Church would, if handling it herself.