Catholics, as far as I understand, were largely unified by papal authority, and the Orthodox by the patriarchate. The plethora of Protestant denominations are an enigma to me, however.
I'm not even concerned with doctrines and practices, or with how religion shaped international relations and politics. What I want to understand is whether the average Calvinist would see Arminians and Zwinglians as heretics or as brothers in faith, and whether an Anglican would consider himself to be closer in kinship to a Lutheran or to a Calvinist, and how remnant Hussites viewed Protestants in general. So basically I'm trying to figure out whether some of these denominations can be grouped together in the sense that the man on the cobbled street would feel a sense of community with others belonging to different denominations.
(Also, is it correct that the Catholic-Protestant schism ceased to be a major factor in most of Europe only around the late 19th century?)
I'll answer your question as far as my knowledge extends, which is with regard to Lutheranism and its relationships with other groups. The differences between Lutheranism and Calvinism were far less pronounced at the beginning. The nature of the Lord's Supper was the primary sticking point, but Calvin was able to subscribe to the Augsburg Confession after some tweaks were made to its Eucharistic language (the Variata of the Augsburg Confession, also the reason why Confessional Lutherans today subscribe to the Unaltered Augsburg Confession).
In 1557, Lutheran theologian Jakob Andreae met with Calvinist theologian Theodore Beza at Göppingen, in an attempt to create a document on the doctrine of the Lord’s Supper upon which both Lutherans and Calvinists could agree. Beza and Andreae actually managed to come to a sufficient consensus to satisfy themselves, but their statement was rejected by both sides and nothing further came of their work together.
Those two met again in 1586 in Montbéliard, France. This city was of particular interest and under special pressure, belonging to the solidly Lutheran Duchy of Württemberg but was sandwiched between the Roman Catholic Lorraine on one side and the Calvinists in the Palatinate and Switzerland on the other. In border areas such as this, even by 1586 it was not impossible to consider oneself simply "Evangelical" without distinguishing between Lutheran and Calvinist. The colloquy itself was sparked by a sudden influx of Calvinist Huguenots into Montbéliard due to a 1585 French decree demanding that they return to the Roman church or leave the country. These exiles asked if they could take the Lord’s Supper in Montbéliard’s Lutheran churches; the following year, the Colloquy of Montbéliard was called to once again attempt to address the perennial sticking point of the precise nature of the Lord’s Supper. Sadly, this time there was no satisfactory understanding to be reached, but it does illustrate that for at least two generations after Luther's death it was possible, at least in certain areas, to not distinguish so sharply between Luther and Calvin.
That's just one example, but the divisions did not develop overnight. As for your question about the average man in the street? There were a number of highly involved laymen in the Reformation; a number of prominent Lutheran theologians were never ordained. The average person listening to sermons in church would probably hear a great deal about the differences between Roman Catholic and Reformation doctrine and practice, as this was a not-insignificant theme of Lutheran preaching, so they'd probably be quite knowledgeable in that... But between Lutheranism and Calvinism, much less so.
Keep in mind that there was no freedom of religion as we imagine it today; a particular ruler's territory would not include churches from multiple denominations. The principle was called cuius regio, eius religio; the religion of the ruler determines the religion of the territory. So one province or principality would be Lutheran, while another would be Roman Catholic, and another (a little later, once it was more well-defined; originally it only recognized Lutheran and Roman Catholic) would be Calvinist. There wouldn't really be a Lutheran church just down the street from a Roman Catholic church within the same city.
Chung-Kim, Esther. Inventing Authority: The Use of the Church Fathers in Reformation Debates over the Eucharist. Waco: Baylor University Press, 2011.
McClean, Charles Louis, Jr. “Theological Issues at the Colloquy of Montbeliard 1586” (S.T.M. thesis, Concordia Seminary, St. Louis, 1974).
Nugent, Donald. Ecumenism in the Age of the Reformation: The Colloquy of Poissy. Cambridge: Harvard University Press, 1974.
Raitt, Jill. “The Emperor and the Exiles: The Clash of Religion and Politics in the Late Sixteenth Century.” Church History, 52:2 (1983), 145-156.