They did! The limes Germanicus ("German boundary") were a series of fortifications, primarily forts and turf embankments with palisades, that were used to "plug" the gaps in the natural barriers of the Roman empire. This was actually quite a bit longer than Hadrian's Wall--the Upper German-Rhaetian Limes is about 550 km as compared to the 135 for Hadrian's Wall, although not continuous. This sort of linear frontier fortification was actually somewhat common in the ancient world, most famously the precursors to the Ming Great Wall in China, but there was also a massive series of fortifications build by Sassanian Persian to plug its northern frontier.
There is the question of why Hadrian's Wall was so grand, however. We need to be a bit careful with our own biases here, it is natural to assume that anything in stone is always better because stone survives better than turf, but Hadrian's Wall does seem to be unusually elaborate. There are, I would say, two decent explanations for this, as well as one bad one that is oddly popular.
The first explanation is pretty simple: the frontier with Caledonia was indeed unusually restive. Even though it does not receive the literary attention or have the ultimate historical importance of the Rhine and Danube frontiers, archaeological surveys have found that it actually has a significantly higher density of fortlets (mini-forts that housed fewer than 100 soldiers). Thus, while the Caledonian frontier did not pose the same level of danger to Rome, it did require an unusual level of military attention, and thus it is not surprising that it received an unusually elaborate fortification. The other possible explanation is that the emperor Hadrian got a bit of a bee in his bonnet about the matter. There are again two solid reasons to think this: one is that the later Antonine Wall was constructed of turf and timber despite being in a comparable location, so it probably was not a matter of necessity or locally available materials. The second is a somewhat amusing bit of evidence, namely that for the earliest phase of constructions the milecastles were spaced to have strictly uniform distances between them rather than to follow considerations of terrain. For later construction, this changed. The argument goes that Hadrian, who fancied himself something of an architect, micromanaged the early construction to its detriment, but when his attention became focused elsewhere the more practical considerations that usually drove Roman fort construction took charge.
This is speculation, but it is solidly founded and points towards considerations beyond the practical involved in the Wall's construction in the way it was. But why would Hadrian take personal interest? There is some very recent scholarship pointing towards a severe bit of unrest in Britain at the beginning of his reign, and that the fire that burned London at that time was not in fact an accident, and was instead connected to some war or rebellion that has been missed by our rather thin sources for the second century. It could be that this was on Hadrian's mind when he toured the province in 122. So the story goes somewhat like this: Hadrian comes to power in 119, and this triggers some sort of major conflagration in Britain. It is put down by the time he gets there, but he decides the way things are do not work, and so orders the construction of a massive wall (which actually does seem to work). Other frontier fortifications were built or elaborated on at the time but were not so closely connected to his shaky ascension, and so received less attention.
Ed: slight alteration, I accidentally conflated two events, detailed here
The bad explanation I mentioned is that the wall was not in fact a Wall, it was a Symbol, and that the main thing it guarded was Ideology. This is a remarkably common explanation despite the fact that there are considerable problems with it (historians, particularly those who do not really focus on military history, have an odd habit of looking at walls and determining that they are a sort of rhetorical device). Firstly, the wall was well away from the major population centers in southern England and the midlands, there was very little reason for most people to see it or have any interaction with it at all. If the reason it was built was more to cow the rebellious Britons than defend the province, the logic of its placement was certainly driven by military matters. The second reason is that Hadrian did not advertise the construction of the wall, he did not strike coins with its image or depict it in any sort of artistic work (like the Dacian forts on Trajan's column). If it was a propaganda device, it was one that was not very widely propagated.
So the quick answer to your question that while the Caledonian frontier was unusual, it was not really all that unusual.
A source I would recommend for this topic is David John Breeze's The Frontiers of Imperial Rome. I am also heavily influenced by Matthew Symonds's Protecting the Roman Empire.
They did! When you're speaking of a border wall in Britain, you're talking about 'Hadrian's Wall' (or maybe the 'Antonine wall' which was constructed further north, but also much later) which is actually a part of a much larger defensive system which we call the 'Limes' (latin for 'border').
In mainland Europe the most famous of these defensive structure is called the 'Limes Germanicus', which was split in two parts and ran through what is today Germany and the Netherlands. It ran along large rivers, which acted as a natural frontier, and consisted of a roughly 500 kilometer long 'wall' with dozens of forts and hundreds of watchtowers alongside. It's important to note though, that we are not talking about a massive stonewall that ran along the entire border - in fact for most of it there was simply a large ditch with wooden stakes as a barrier (similar to what you might expect from a roman camp). Only in select parts, where an enemy might cross the natural (river) border in significant strength with relative ease, where additional precautions taken. At those points you might find stronger wooden (or even stone) walls, with additional fortifications where troops could be stationed.
Unlike its British counterpart however, not much is left of the Limes Germanicus, which you would today immediately recognise as a fortification, which makes sense as wood rots and a ditch is, well just some earth thrown up on a pile. Unless you know what you're looking for, you probably wouldn't notice it. Nowadays, several reconstructed forts and watchtowers exist, which provide a great insight as to how the entire fort-system might have looked. A great example of this is the Castellum Saalburg, a reconstructed stone-fort in Hessia, Germany.
Just in case you're wondering why they didn't build a wall of stone/rock as they did in GB: There are two major reasons. Firstly, Geography. Much of the Limes Germanicus ran through massive forests and hilly areas, where it would have been incredibly difficult to construct such a wall. Secondly, cost. Going of the difficulty of the terrain, adding the length of the border and the manpower needed to construct such a wall, nevermind quarry all that rock - the cost would have been ruinous. And as it was mainly build to discourage raiding partys, and not full-scale invasions, the cost-benefit made it unnecessary.
There were also several other border fortifications throughout the Roman Empire, which we today classify as part of the 'Limes'. Remnants of them and arcaeological sites can be found in Libiya, Jordan and along the Danube all the way to the black sea.
Sources:
The Roman Frontier in Germany: an Archaeological Survey - H. Schönberger (The Journal of Roman Studies , Volume 59 , November 1969 , pp. 144 - 197).
Frontiers of the Roman Empire - Hugh Elton (1996).
Only in Britain do we see a permanent, monumental stone wall, running for 80 miles from the mouth of the River Tyne near modern day Newcastle to the opposite coast. But even this was not the only method of protecting the British frontier: there was nothing but posts and forts until construction of Hadrian's Wall began in 122 AD (the Stanegate frontier), and from 142-161 the frontier was moved north to the so-called Antonine Wall, a system of ditches, a linear turf embankments (albeit with a stone foundation) interspersed with posts and forts. Still, Hadrian's Wall forms one of the most iconic of Roman frontiers, simultaneously an "argument in stone," as well as a formidable barrier to movement. But it also made sense given the relatively narrow "choke point" that is northern Britain in general, where one long stretch of wall could reasonably hope to control the entire 80 mile frontier. But other frontiers were much longer: the Danube runs for 1771 miles!
It should be noted while we don't see long stretches of monumental stone walls elsewhere, we do see other long linear artificial fortification systems elsewhere. Most notably, the Agri Decumates, the stretch of terrain between the Rhine and Danube, was fortified by a system of ditches, palisades, watchtowers, and even stretches of stone wall. Supported by roads, posts and forts, this represented a complex and sophisticated defensive system.
In Africa, a series of ditches (fossae), extending for roughly 450 miles, were constructed, sometimes backed with earthen ramparts or even small stone walls, used to control the movements of nomadic peoples ; the goal (as with all Roman frontiers) was less an airtight preclusive defense, and more a system that channeled movement and facilitated surveillance.
But in most instances, however, the Romans leveraged riverine frontiers: the Rhine, Danube and Upper Euphrates. Rivers were quite useful, as they at once provided a serious barrier against opposing peoples, allowed for Roman riverine patrols to efficiently monitor the frontier, and also facilitated the movement of troops and supplies laterally by river to different parts of the frontier (supply ships and patrol boats can be seen, for example, on Trajan's column supporting his trans-Danubian invasion). The river makes a continuous, expensive stone wall unnecessary, although the river barrier was supplemented by a defense in depth system of auxiliary posts, forts and legions (as was, of course, Hadrian's Wall and all artificial frontiers).