No. The Romans weren't alone in equipping common farmers as effective infantry. This has been as common a tradition as using nobles as effective cavalry.

Classical history is replete with instances of conscripted farmers being used relatively effectively in the field. The ancient Greek city states had farmers arm themselves and fight in neighborhood units, while the Spartan and Macedonian militaries state-armed common farmers as well. The Gauls and Germans who fought the Roman armies were often primarily peasant troops. All the aforementioned units made tactical use of the phalanx, as did the Swiss pikemen hundreds of years later. What must be emphasized is that the phalanx as a system is not very unique. Art relics show that the Egyptians and Sumerians made use of massed infantry formations. Much of the fighting efficacy of the Roman military came not only from professionalism, but also good generalship, logistics, manpower, and other soft-power realities.

It is not even true that cavalry arms were always aristocratic in character. In fact, this stereotype comes from the time of Aristotle and the late-Platonic theorists. As a cliche it holds a certain truth: in high-density farming communities, it's relatively difficult to sustain much more than a mounted elite, and counterproductive after a certain point to raise cavalry while dispossessing farmers who would often gladly fight to defend a more moderate political regime. However, nomadic tribes like the Arabs, Scythians, Mongols, Huns, or Persians had a vast surfeit of low quality prairie land, enabling thousands of low-ranked tribal members (the equivalent of peasants in settled societies) to serve in a mounted role. In low quality brush land, farming was less sustainable than herding. While such tribes rarely could field large mounted contingents for over a season in the wrong climes such as northern Europe, and often converted themselves into elites of conquered farm land (think the Bulgars and the early Goths), it is nonetheless important to emphasize the relatively common background of many of these light cavalry fighters.

Slave armies should not be ignored either. While I admit that it stretches your definition, the effectiveness of Janissaries, Mamelukes, and Ghilman should not be ignored simply because they were enslaved by the state. As children, members of these fighting castes came from the common peasantry and they were trained, equipped and organized as effective professional soldiers ruling for centuries over their respective regions. Their status as slaves moreover should not truly distinguish themselves from other impressed, professional armies of various other empires, such as the Russian, Incan, Chinese or early modern European. These militaries conscripted troops for life or 20+ years. Official enslavement seems a matter of semantics under such brutal circumstances. As professionals, such troops were man to man often as deadly as any Roman legionnaire.

As to any unique professionalism, the Romans scarcely had any monopoly. They weren't invincible: Romans well understood their own limitations and authors like Livy were even able to turn this into something glorious: stoic triumph despite adversity. What the Romans had in addition to discipline was staying power and stubbornness. Remember, Rome was one of the westernmost city-states in the world of its era, and was thus capable of mobilizing immense resources from the hinterlands when compared to states like Athens or Pyrrhus. It is true that the Republican and Imperial Rome had high morale, strong patriotism, often more troops than their neighboring rivals, and, at least in the beginning, a political constitution that created a helpful stability enabling Romans to weather such defeats as they faced.

Goldsworthy - Caesar; Why Rome Fell

Livy - The Punic Wars

Seutonius - Lives of the Caesars

Kagan - The Peloponnesian Wars

Holland - Rubicon

Dando-Collins - The Tenth Legion; The First Legion

O'Shea - Sea of Faith

Norwich - Byzantium

Bicheno - Crescent and Cross

Well this question hinges on which Rome: Republic or Imperial? Looking at the Republic (well up to the Late Republic) the Roman army was a militia composed of landowners who met a certain property requirement - no different from any other Greek city state. So 'lads from common backgrounds' wouldn't be the poor plebs but a bunch of relatively well-off 'middling' class.

A differing factor was the level of discipline imposed within the army. Roman discipline was notorious and a point of pride for the Romans themselves. Polybius emphasizes this when he explains how the Romans build marching camps every day and to the same exact plan. In contrast Greek armies were more of a confused sprawl and commanders (at times) had difficulties controlling their men outside of battle. A Roman general (one with imperium) had the authority to execute citizens - a power a magistrate did not have normally.

By the Late Republic you no longer had a civic militia but in effect a volunteer force bound to their commanding general. These served in multi-year campaigns in a variety of theatres and you can see specific tactical employment of newer legions (e.g. Caesar consistently keeps them in the rear or 2nd line with his veterans forming the battle line). This allowed more time for newer recruits to learn the drill, the tempo of army life, to build up unit cohesion. This was something not available to new recruits in the early to middle Republic where armies would be formed in response to wars (usually short single seasons) and they'd be formed up and marched off to battle - it was noteworthy when a general took the time to drill his army prior.

Obviously with the Principate you get the regularization of the army and it became fully professional.