I don't have any concrete statement from Gen Urobuchi or the anime production staff, but I suspect it's because although Alexander the Great began as King of Macedonia, he spent a good portion of his life ruling Persia and other Middle Eastern kingdoms, and so was actually more of a Middle Eastern king than a Greek one.
According to the Wikipedia article, Alexander reigned from 336 BC to 323 BC—a reign of only thirteen years. He spent two years as King of Macedonia, rallying his forces for an invasion of Asia. In 332 BC, just two years after beginning his conquest, he overthrew Darius III, Pharaoh of Egypt and King of the Achaemenid Empire in Persia, and assumed those titles. That means that, of thirteen years of rule, he spent nine as King of Persia and Macedonia and just two as King of Macedonia; furthermore, we can see from the Wikipedia page that he spent almost all of his time after the invasion in Persia and the Middle East. As @Memor-X mentions in his answer, the Achaemenid Empire was one of the largest empires in the world by population, so being its king put Alexander over a far greater number of people than his kingship of Macedonia. As Memor-X mentions, "Iskandar" is Alexander's name in Old Persian, and all of these new subjects would have known him by that name. And after taking the kingship of Persia, Alexander didn't double back to Europe; he continued east, pushing through Pakistan and into India, searching for the great ocean at the end of the world (as his fictionalized counterpart Rider also was), planning to establish a capital at Babylon and invade the Arabian Peninsula. Since all the people on his eastward march would be more familiar with Persian than with Greek, it makes sense that they would also call him "Iskandar".
Not only that, but during his time as ruler of Persia, Alexander adopted several Persian customs and overall tried to integrate his Macedonian and Persian subjects (Source, the first paragraph, and also here.) Alexander also married two Persian princesses, Stateira II and her cousin Parysatis II, and integrated his army so that both Macedonians and Persians could hold positions of high power; he bribed, pandered, and punished as needed to keep the peace between them, even having some of his men executed for desecrating the tomb of Persian king Cyrus the Great.
Given all that, it seems possible that Urobuchi thought that even though Alexander began as a Greek king, his major accomplishments were more associated with Persia and the Middle East, and so it was the Persian form of his name that should go down in legend. His Noble Phantasm, Ionioi Hetarioi, transports everyone to the Persian desert, not the Macedonian hills, which could be another nod to this. It's also notable that, so far as I can remember, we first hear the name "Iskandar" from Rider himself, in Episode 4 when he flies into the middle of Saber and Lancer's battle and tries to recruit them for his army; perhaps Urobuchi thought that, given all the history that Memor-X and I mentioned, Rider would think of himself as Iskandar and not Alexander.