There’s a few different aspects of this to consider when trying to understand its significance.
First is the obvious symbolism of the action at this particular place and moment in time. Whether one shares the view or not, the reality of the current situation is that too many people in today’s world view Christian-Western/Muslim relations through the prism of the Clash of Civilizations, an ongoing confrontation which, for those inclined, dates back to the advent of Islam. The battle over Constantinople/Istanbul has been a key symbol of this supposed Clash. Much of Erdogan’s rhetoric and action suggests this is the basic historical framework by which he understands his role as the President of modern Turkey, and it would be naive to think it hasn’t played into his calculations. The building was the greatest Church on earth for close to 1,000 years at the heart of the greatest Christian empire and successor to (or continuation of) the Roman Empire. So this move has been made in the knowledge that it will be viewed by many as an act of aggression/provocation or triumph, depending on which ‘side’ of the Clash one stands.
Second is the precedent it draws on and sets in terms of analogous disputes over religious sites globally. While each case has its own historical and contemporary dynamic, there are obvious parallels to be drawn with sites such as those at Ayodhya, Córdoba, and Jerusalem/Hebron. I wonder if/how much the recent Ayodhya ruling impacted Erdogan’s decision here? Perhaps very little. He can also point to Córdoba for precedence. On the other hand, I find it hard to believe he hasn’t considered the potential impact of this on Jerusalem, where Jewish fundamentalists will surely be thinking in terms of how his justification may be used in their campaign for a third Jewish temple on the Temple Mount, perhaps in place of the al-Aqsa Mosque and Dome of the Rock. Which means that he has determined that the inevitable fall-out is either worth it or desirable.
Finally there is the question of how it relates to the unresolved legacies of Ottoman rule in the Balkans and Anatolia. The last century or so of Ottoman rule in/withdrawal from these regions involved ethnic cleansing of both Muslims and Christians on a mass scale, and concurrent mutual destruction of religious sites. The cleansing of Christians from Anatolia, especially during and after WW1, is by now quite well known in the West and often used as a stick to beat modern Turkey with. What is much less known about in the West is the expulsion of Muslims from the Caucasus/Crimea and the Balkans due to Russian expansion in the former and national independence movements in the latter. These refugees ended up sprawled across Anatolia and Ottoman Syria, and their descendants make up a considerable percentage of the population of modern-day Turkey. Yet there is virtually no recognition of their plight in Europe today, or of the fate of their mosques in places like Greece. We know for certain that this aspect played into Erdogan’s thinking, as he has explicitly referred to it. Of course Turkey’s own intransigent position on the Armenian genocide has been perhaps the major obstacle to facing these legacies, but perhaps if there had been a genuine campaign for mutual recognition of the suffering endured by all the peoples of these regions during the collapse of the Ottoman Empire, modern day Turks would be more receptive to a dialogue based on mutual respect for the conflicting religious traditions.