When it comes to the history of political ideas and thought a lot of terms and conceptual pairs - such as "left/right" in this case - are very hard to define in a clear and concise way. The fact that even or rather especially in today's political scene these distinctions into "left" and "right-wingers" are used pretty much worldwide to describe (at times) completely different things (people, parties etc.) - often in a polemic or oppositely a proud manner - underlines the relevance of this theoretical backdrop. As with most of these historical terms they have their very own history that we need to be aware of at all times. In short, the german philosopher Hans-Georg Gadamer once described it as follows (it should be noted that the way Gadamer used the term "historicity" itself is disputed, most notably by Theodor W. Adorno; rightfully so one could [as I would] argue but this is not of primary importance here):
"Real historical thinking must take account of its own historicity. Only then will it cease to chase the phantom of a historical object that is the object of progressive research, and learn to view the object as the counterpart of itself and hence understand both. The true historical object is not an object at all, but the unity of the one and the other, a relationship that constitutes both the reality of history and the reality of historical understanding. […] I shall refer to this as “history of effect”. Understanding is, essentially, a historically effected event." (Gadamer, Hans-Georg: Truth and Method. (First published 1960). London 2013. p. 310).
Returning from this theoretical meta-level which I just thought was important enough to mention, let's take a short look into where the usage of the terms "left" and "right" in a political sense stems from. Initially, we find a relatively clear-cut origin: The ever-interesting French Revolution. In the "Assemblée nationale constituante" or the succeeding "Assemblée législative", a national assembly-like body formed in 1789 (subject to several reforms) the nobility/aristocracy, the main group of supportes of the king (called "les Monarchiens"), claimed their place in the assembly to the right of the chair occupied obviously by the king just as Jesus who is "seated at the right hand of God" (there are hundreds of similar descriptions of that in the Bible), thus alluring to the idea of "god-given monarchy". Enemies of the crown were therefore sitting on the left hand side of the chair during proceeding. By that, the strong conflict about the envisioned system of state and government was effectively visualized and represented by the actual seating arrangements. This was later copied in other, sometimes newly-formed parliaments in Europe, such as the "Frankfurter Nationalversammlung" during the revolution(s) of 1848 (the year Marx and Engels published the "Communist Manifesto").
By that time, roughly two decades before in the early 1830s, this distinction into "left" and "right" had in some parts at least already left the purely parliamentary-political context and was applied also in philosophical debates - naturally as way to express a stark contrast to the other group; I'm alluding first and foremost to the arguing followers of the german philosopher Hegel (who had died 10 years earlier) that in this process split up into the "Old" or also "Right Heglians" on one side and the "Young" or "Left Heglians" on the other. Marx and Engels can be counted to the latter. Among others, the arguments about criticism of the existing conditions and systems of governemnt, statehood, society and religion (Feurbach for example was also a "Left Heglian") illustrated the contrasting positions and understanding/interpretation and continuations of Hegel's work, especially concerning the concept of "Dialectics". Marx famously described himself in his preface to the second edition of "Das Kapital" as having "turned" or "stood Hegel on his head" although that's a question that could be covered in its own thread, and probably already has been better than I ever could.
At last, it should be noted that for the longest part of their lives neither Marx nor Engels used the word "left" in a self-descriptive way. To them, "left" and "right" was still mostly a distinction confined to the matters of parliaments in which their own movement did not want to play a role - which later lead to further fragmentation within Marxism when the so called "reformists" started to criticize the "orthodox" Marxists in the 1890s and proposed the use of lawful means, only lawful means to achieve and initiate the social change envisioned by Marx. In this sense, the evolution of the communist movement into mass-parties of socialists and workers and the establishment of the First International automatically brought it closer to the parliamentary cosmos/sphere.
Long story short, the use of the term "left-wing" before Marx and Engels was heavily shaped by the events in the National assembly during the French Revolution when the distinction was bound to the respective opinion about the system of state between republicans and monarchists, which evolved to progressives and conservatives. Thus, "left" could mean a variety of things from liberalism to the ideas of enlightenment and the will for political representation and self-governing. This way of political labeling was then further reinforced in other parliaments across Europe in the 19th century - the role of the revolution(s) in 1848 and following can not be underestimated in their influence on "left-wingers" and altering their self-image. As to why this conceptual pair seems to have existed throughout very different times, some of it might be down to the easy and illustrative way it depicts the political spectrum that gives a continued frame of reference. As I said though, Marx himself did not initially see communism as part that frame. The term was always bound to its time, just as it is today.
Sources:
Haug, Wolfgang Fritz, Frigga Haug, Peter Jehle, and Wolfgang Küttler (Eds) (2015) Historisch-kritisches Wörterbuch des Marxismus, Band 8/II ("links/rechts bis Maschinerie"), Hamburg 2015 [Historical-Critical Dictionary of Marxism, Vol. 8/II ("left/right - machinery")], 1153-1168.
Bobbio, Noberto: Left and Right: The Significance of a Political Distinction. Chicago 1996.
Fuhse, Jan A.: Links oder rechts oder ganz woanders? Zur Konstruktion der politischen Landschaft. In: Österreichische Zeitschrift für Politikwissenschaft. Bd. 33. Nr. 2. 2004. S. 209–226. [German]
1911 Encyclopædia Britannica, Volume 11. French Revolution, The by Francis Charles Montague and Arthur William Holland.
Marx, Karl: Capital: A Critique of Political Economy. (Edition from 1906).
Edit: Corrected some or admittedly rather quite a lot of grammatic mistakes etc. I wrote this comment late at night is all I can say in my excuse. If you see any more glaring mistakes or have other problems, please voice them!