I struggle with whether we truly have the same exact faith given that a lot lies in the details. I don't have the theological acumen to sort it out, but it seems to me that if we really had the same faith and that history and misunderstanding was the actual problem, we'd have united by now. After all, there have been many prior attempts to do so.
These prior attempts should not be ignored, but they seem to stem from the same reasons some people presently are hesitant to accept a unity. That is, how could we unite without showing that our forefathers were wrong? Sounds easy to answer today, but not then, especially when much blood was spilt over this. Because I don't really think it's primarily about their "they knew what they're talking about" as it is much about "they suffered for the faith". If you want to latch on to the strongest reason why one shouldn't unite with the other, it is because, as one might imagine would say to the other, "You murdered our saints!" It was said of St. Dioscorus concerning Flavian's beating (who as I understand, incidentally, his commemoration in the EO Church was yesterday) and turned it as an emotionally charged issue to try to cast him out. It was said of Leo concerning St. Dioscorus' beating, and it became an emotionally charged issue of fighting tooth and nail against the council. It was said of the anti-Chalcedonians killing Proterius, and it was said of the Chalcedonian emperors killing anti-Chalcedonians (we just celebrated the memory of a saint today, St. Barsouma, who was said to have been tortured for rejecting the council, and departed only 7 years after the council). Monks were also involved in killings, whether in Palestine, Syria, or Egypt, Chalcedonian and non-Chalcedonian. It was a sad time for Christianity. Zeal became violent, and the clarity of rational discussions were stained with the blood of their loved ones. Chalcedonian patriarch Timothy III Salophakiolus even seemed saddened at the whole prospect, seeing it as "Christians killing Christians".
We have a belief in Egypt: The killing of sons by the government produces a need for a revenge by the mothers stronger than the need of religion. (I think the saying goes: A mother's vengeance for her son's death is her religion.) How much more true it is anywhere when the shedding of blood happens either way, causing a cycle of vengeance, and if no vengeance, a stubbornness to conform.
So, why exactly has the notion of unity became so strong today? Because it probably took more than a millennium of Islamic oppression to put us back to our senses, to contemplate our past actions, and to came back in the age of world-connectedness, with full freedom, and no fear or pressure from cultural or political threats. The past attempts showed us the wrong way to have discussions. Today, there is a real chance to openly question each other with full respect and no verbal attacks what exactly our respective fathers believed and how this can be interpreted in light of today.