I'm going to take a stab at explaining what the inquirer is asking about in the original post. First, the Eastern Orthodox Church and the so called Oriental Orthodox Churches, do not at present have a canonical relationship, mostly due to differing understandings of doctrinal issues debated during the middle of the 5th century. These differences are explained in a link above.
Since its inception, the Eastern Orthodox Christian Church has maintained the administrative tradition of the early Apostolically established churches, which became known as the Pentarchy, the Ancient Patriarchates of the Roman Empire, led by the Church of Rome, the capital of the Empire; two other provincial capitals, Alexandria, (Egypt); Antioch; then Jerusalem; and since 335, New Rome, Constantinople. It can be considered as a model of local control. Bishops held exclusive authority over teaching in their diocese. Bishops (or archbishops, if they were in larger cities) were bound together by Provincial (Eparchial) Synods for matters of common concern, which were under the authority of the Holy Synods of these Patriarchates. Matters of doctrine and other matters of common concern which may have been in dispute, were addressed by Ecumenical Synods (Councils), assemblies of all the hierarchy of the church. None of the churches could interfere in the affairs of the other patriarchates, except appeals of substantial matters in dispute could be heard by Rome and later Constantinople. This is the essential pattern of governance, or church order, observed even today by the Eastern Orthodox Christan Churches.
As the church expanded into central Europe and later Ukraine and Russia, a church would become autocephalos, the ability to elect its own primate or first hierarch (and all its hierarchs) and it was no longer subject to the authority of its Mother Church. Give this tradition of internal governance, some practices developed in isolation within each of these churches, that today are viewed as cultural variances, but all the churches hold to the same doctrine of Faith and share Holy Communion.
Church unity is expressed in many ways. One of them is heard when a primate of an autocephalos church celebrates a Divine Liturgy. At the petition, "Among the first, be mindful Lord...," where the bishop is commemorated, when a primate of an autocephelos church celebrates, he (or his archdeacon) commemorates each of his fellow primates, likewise, so does the Ecumenical Patriarch of Constantinople, the "First Among Equals." Also, when a primate is newly enthroned, he will visit all of the autocephelos churches, and the primates of those churches, will later return a reciprocal visit, concelebrating the Divine Liturgy typically.
Fr. John Meyendorff has written that the appellations "Greek," Bulgarian," "Romanian," etc. began in the churches which were under the yoke of the Ottoman Empire, during the 18th century, when revolutionary movements for the independence of these provinces were forming. He noted that even in these lands, in their native languages, when someone was asked, to which religion do you belong, the response was, "I am Orthodox." Note too, liturgically, you will never see the ethnic appelation used in reference to the church, "Again we pray for the pious and Orthodox Christians."
On the First Sunday of the Great 40 Day Lenten Fast, the Sunday of Orthodoxy, the same affirmation of Faith (Synodikon) from the 7th Ecumenical Synod (787), is read in all the churches worldwide, "...as Christ has awarded...thus we declare...This is the Faith of the Apostles, this is the Faith of the Fathers, this is the Faith of the Orthodox..."
My Reply specific to the American ecclesial phenomenon follows below.