Preface: I hope this post doesn't come off as too rambling. It's basically a thought experiment that developed as I wrote. Just scroll down for my tl;dr if you like...

I think this is a fantastic question, very complicated, with so many ways to look at it, and one could easily come down on either side of the fence. In this case, having read the good comments in the thread you linked, I will define the question as "did borders exist before the Renaissance," the fence representing answers to this in the affirmative or negative. I won't go into whether it was only codified in the Peace of Westphalia (1648), that's not my area and I bet that's something else you could argue all day.*

I note that the person who produced the quote in the OP said that the Roman Empire was the "exception [rather] than the rule." So he seems to accept, for good or for ill, that the Roman Empire did have borders and yet other states/tribal groups did not. Some posters in the thread point out quite rightly that some Roman borders could be considered to be more amorphous than a well-defined line on the map that was or was not Roman territory, but I personally think Roman territory was pretty clearly defined from an administrative: it was their sphere of influence which was not so clear. That is to say, the Rhine and Danube may have been Roman borders, but if Rome saw something it didn't like in the other side of those rivers, then through diplomacy or force they could make things happen the way they wanted it. An example of this cross-border power relationship between Rome and significant tribal groupings of the Rhine and Danube is the planned campaign of Tiberius against Maroboduus in AD 6, which must have made the latter very nervous, interrupted by the Great Revolt in Illyria. And yet, when Varus and his three legions were destroyed in AD 9, and the head of Varus allegedly sent to Maroboduus, the latter sent the head back to Rome for honourable burial. Clearly, even though Maroboduus was in a much better position than he had been a few years earlier, he was not interested in provoking Rome any more: he had found a status quo he was happy with. Obviously, this relationship between the Romans and the Marcomanni, who became an even more significant tribal group over time, broke down under Marcus Aurelius, and Rome had to resort to significant campaigning in the Rhine and Danube area once more to enforce its will. All this time (nearly two centuries, with many events omitted), the Rhine and Danube were the "borders" of Roman administrative control, but the physical battlegrounds were on either side of that border at different times, and the issue at hand was not the integrity of the border itself but Rome's domination in its sphere of influence (and direct interest), which obviously extended beyond the border.

The Rhine and Danube rivers appear to have been administrative borders, and I could not even begin to summarise the endless debate over whether or not they were chosen for their strategic advantage as natural barriers (a logical assumption) or for administrative simplicity at the edge of the empire's advance, but here is something very important to note: The Roman Empire's borders served different purposes in different places, in different ways. So while Hadrian's Wall was a man-made fortification (due to its size and insignificant height probably a symbolic border, but the presence of significant military forts built into it also clearly betray the role of military control), and the Rhine and Danube were rather natural barriers, we have evidence of ditches and other man-made barriers in Africa, and yet no such thing exists in the East with Persia, where fortified cities rather served as centres of control for both Roman and Persian. The Romans may not have had any sort of maps as we understand them today (Isaac 166), and so it is very unlikely they saw the Empire has a set of clearly drawn lines or "borders" that could not be crossed by the enemy. And of course even if such lines were drawn on some map, they would be meaningless: the Rhine and Danube, Hadrian's wall, the trenches and army bases in Africa, and the total lack of any such linear thing in the East as opposed to various fortified cities should show that such a line could not serve the purpose it did later in Europe, where a line could be clearly drawn on a map and you were doing the wrong thing by crossing it under arms.

So did the Romans have borders? What is a border anyway? If a border is a line drawn on a map that can't be crossed without being in enemy territory or whatever, then it is an incredibly abstract thing indeed. And borders took different forms in different places: the Romans did not have walls along the entire frontier of the Empire, as I have suggested. A border was and is, in my view, as much symbolic and moral than physical. "Barbarians" crossed Hadrian's wall or the Rhine and Danube or walked past fortified cities in Syria all the time: there does appear to have been a great deal of freedom of movement in this regard. But a military force crossing into Roman administrative territory was a different thing: the Romans would use their military to oppose such incursions, and they would enter "enemy" territory to extract penance for such conduct. So the question of violating the "border" is one as much of intent as physical movement. So if a border was not a clearly defined line on the map, but rather a very physical thing such as Hadrian's wall, the Rhine and Danube, or the fortified cities of the East, its violation was far more abstract.

So really I might have actually contradicted myself here, as in the footnote I typed before the rest of this post, I stated the Peace of Westphalia established an abstract principle for the maintenance of borders that were physically well defined. I guess in that regard, the true "border" in the modern sense could only be established once the abstract understanding that physical boundaries and lines cannot be violated is made clear. That said, I think as the Romans did have borders, and the person voicing the quote in the OP even seems to admit that, even with qualification, I don't think it can be right to say it's a "relatively new" concept. Treaty of Westphalia or not, the Romans had a good understanding of their administrative territory, and they had an intensely keen sense of when these borders were being violated, or Roman power was being disrespected. Considering the true importance of the border was a more moral and abstract value as representation of a state's power, and the physical line of the border was actually ephemeral to these issues of hegemony, honour and sovereignty, I would suggest that borders are a very old concept indeed. A tribal group might not be able to point on a map what territory was theirs and what was not, but they would have a very keen sense of when their sovereignty was being violated. Such misunderstandings, open to manipulation, were quite typical of Rome's relationships with its neighbours in the Roman Monarchy and Early Republic, where any movement of armed men could be a pretext for war and almost any incursion could be justified as pre-emptive or punitive. So I wouldn't say borders are new, I would just say that eventually states developed a sophisticated enough way of defining abstract ideas such as honour and sovereignty within a physical space, allowing interactions between states to operate off of a clearer set of rules, setting the stage for a world in which offensive and defensive behaviour is easier to define.

TL;DR: I think borders of some description have been around a long time, and Rome is not the only example. If a border is not necessarily a physical line, but a more abstract understanding of what is your city/tribe/empire's territory/area of influence, and what is not, I think that is a concept that would have been understood by many different groups over much of human history. Such an understanding would have been central to a city/tribe/empire's views of sovereignty and honour, very old concepts indeed.

*It's my understanding that the Peace of Westphalia established a more abstract principle of non-intervention in the affairs of other states rather than the actual definition of borders per-se, but I'm hardly well informed on this issue.

Sources: The issue of Roman borders is linked with the question of whether or not Rome possessed a "Grand Strategy" in which the Emperor had clear methods and aims for the maintenance of the Empire's power manifesting in a long-term border and foreign policy. It's a very detailed and sometimes laborious debate that can't be avoided on the discussion of Roman borders, but I won't link too many such works. Here's a somewhat random selection:

Philip Freeman, "The annexation of Arabia and imperial Grand Strategy", from The Roman Army in the East, 91-118.

Benjamin Isaac, "Eusebius and the Geography of the Roman Provinces" from The Roman Army in the East, 153-167.

J.E. Lendon, "Primitivism and Foreign Relations", The Classical Journal, 97, 4, 2002, 375-384.

Edward Luttwak, "The Grand Strategy of the Roman Empire; From the First Century A.D. to the Third," Baltimore: John Hopkins University Press, 1976.

Sara E. Phang, "New Approaches to the Roman Army," in Recent Directions in the Military History of the Ancient World, Publications of the Association of Ancient Historians 10, Claremont: Regina Books, 2011, see esp. 112.

C.R. Whittaker, "Where are the frontiers now?" from The Roman Army in the East, 25-41.

Borders as a general concept were most certainly not 'new' and only first seen during the Renaissance.

The poster most likely wanted to allude to how borders were often quite ambiguous, fluid, often disagreed upon, as well as waxing and waning with influence, military strength, and ability to properly maintain/control borders.

Some of these issues became more regulated and ’set in stone’ as we approach the modern era, but some of these issues are still relevant to us even today.

There are many examples of strict control and dictation of borders throughout history long before the Renaissance as well as the fact that historical cartographers were often prized politically because of the importance given to maps and supposed lines on paper that represented the domains of rulers.

That being said, there are many points in history in nearly all places where there were times that where borders lay or who controlled what territory were incredibly nebulous, which of course lead to border disputes.

But to say that this in turn means that borders as we know them were entirely non existent is going a bit far.

All this being said, even in modern times, people’s concepts of what a border really means is often times interesting.

Borders are at once, more porous and more strict than what many imagine them to be.

When the idea of borders is brought up, one inevitably thinks of land, resources, sphere of influence, as well as the movement of goods and people.

In older days, borders were often decided by tradition and historical precedent, much like they are today. Think of counties, duchies, provinces, etc. Think of the provincial daimyo of Japan, or the prefectures of China.

Many of these areas of land were dictated, not by the reality on the ground, but by abstract concepts like claims, authority to rule a specific territory, titles, etc.

The idea that a border could be violated at will if the de jure ruler was weak and unable to enforce his claims is not something that changed significantly during the Renaissance, nor today.

The idea that tribal peoples or other groups with less centralized social/political structure did not know where their claims on territory ended is also misleading.

Even with nomadic groups, like the people who roamed the steppes of Central Asia, territory was a complicated mesh of who was married to who and thus allowed to access certain grazing lands, sources of water, exact tolls on certain roads, etc. Just because they did not think of them in terms of lines on a map does not mean that these people did not have a very concrete idea of where their claims/privileges began and ended.

There were groups that indeed did not think of land/territory in terms of borders or certain stretches of land belonging to one particular party or another, but this again wasn’t a thing that particularly changed with the Renaissance.

I'll tack on to the conversation related to the Treaty of Westphalia (ToW).

It's important to remember that ToW was a response to the 30 years wars and those specific circumstances, not to what existed 1000+ years before in the Roman empire. They established a specific concept of predefined territorial control (i.e. borders) that helped leaders overcome the jumbled mess of conflicting religion obligations that governed the 30 years war. It certainly wasn't the first notion ever, as PapiriusCursor discusses in greater detail, but it is a concept that is still prevalent in how people think about international relations today.