It existed to one of the highest extents in military history. This was one of the things that largely kept morale in the battlefield, to almost a religious extent. Even auxiliary units did, as well. I hope to answer a lot of your questions and pose examples for each-- please reply if I miss anything.

The Eagle

Firstly, as I have mentioned in one of my previous answers on this subreddit, following Marius’ reorganization of troops in preparation for conflict with the Celts, eagle standards were given out to eagle standard bearers, usually containing the unit’s name and number. For instance, let’s take for example Legio X Fretensis, or ‘tenth legion of the strait,’ would have an eagle standard with cloth below it resembling this. The soldiers took great pride in not only their name, but military accomplishments. Their name would be remembered as a tribute to Caesar’s legendary tenth legion, and likewise their heroic acts during Octavian’s war against Antony.

Even auxiliary units, as you mention have similar titles to the actual legions. After all, these people are fighting together for a very long time. A certain permanence would eventually bear its wings in any military unit, so names were given to the auxiliaries as well. For instance, VII Claudia pia fidelis, or XI Claudia pia fidelis both were given names for their heroic acts in the late republic. (Loyal and true = pia fidelis.)

For instance, Caesar’s tenth legion is described to have shown great bravery during the Gallic Wars, such as their decisive charge against the Nervii in their rear in support of Legio XII Victrix and VII Victrix, following the sight of the two aforementioned legions beginning to route. Caesar describes this in his commentaries of the Gallic War, when he says:

“By their [the Tenth Legion] arrival, so great a change of matters was made, that our men, even those who had fallen down exhausted with wounds, leaned on their shields, and renewed the fight: then the camp-retainers, though unarmed, seeing the enemy completely dismayed, attacked [them though] armed; the horsemen too, that they might by their valor blot the disgrace of their flight, thrust themselves before the legionary soldiers in all parts of the battle. But the enemy, even in the last hope of safety, displayed such great courage, that when the foremost of them had fallen, the next stood upon them prostrate, and fought from their bodies; when these were overthrown, and their corpses heaped up together, those who survived cast their weapons against our men [thence], as from a mound, and returned our darts which had fallen short between [the armies]; so that it ought not to be concluded, that men of such great courage had injudiciously dared to pass a very broad river, ascend very high banks, and come up to a very disadvantageous place; since their greatness of spirit had rendered these actions easy, although in themselves very difficult.”

Later in the life of Julius Caesar, this same legion was ordered to retreat at the battle of Dyrrhachium, and with their retreat, accidentally came other legions to retreat in kind. As this betrayed the courage shown by their legion previously, they demanded Caesar decimate them for cowardice. Although he refused, to keep the men’s morale from dropping due to disgracing their eagle, he demoted their standard bearers.

Crowns of Gold

With the names, come smaller individual units that would have the same form of pride as you ask.

Some say that the smallest unit, the contubernium, even had small traditions to them, such as pack animals being deemed as mascots.

Furthermore, during battle, soldiers would be given awards if they have fought ‘pluckily,’ such as battlement crowns, mural crowns, crowns of the preservers, naval crowns, and many more. Polybius says this pretty well in his histories,

"After a battle in which some of them have distinguished themselves, the general calls an assembly of the troops, and bringing forward those whom he considers to have displayed conspicuous valour, first of all speaks in laudatory terms of the courageous deeds of each and of anything else in their previous conduct which deserves commendation”

Furthermore, I will explain what each award took to be awarded, and what they meant to the people it’s given to. Firstly, the battlement crown was a very, very, ornate crown made of gold that was awarded to the first soldier to break the threshold into the enemy camp, or town. This provided a substantial amount of morale to troops when charging a city, to perhaps get this award. Much like the other ones I’m mentioning, it has a hugely symbolic benefit to your contubernium and century, a great honor would be bestowed upon you and your companions. Mural crowns were also very, very expensive and ornate. According to Polybius, these awards were given to men who climbed the walls of the enemy city and planted the century’s banner upon the walls. It’s very common that centuries would often compete to receive awards like these, or face mockery.

Further, the crown of the preserver was awarded to soldiers that saved the life of any citizen or any ally of the empire. Often, it’d be very highly regarded by civilians and soldiers alike, when he says:

“The man thus preserved also reverences his preserver as a father through his life, and must treat him in every way like a parent. ”

If you couldn’t tell by now, these awards have amazing methods of getting soldiers to face danger headon.

Esprit de Corps of Leadership Roles

The structure used following the Marian reforms actually sheds some light on good points about how soldiers were encouraged to strive for leadership roles, such as the primus pilus, or first spear. The primus pilus would lead the first cohort, (an honor in it of itself to even be in the first cohort) and in turn, lead every other cohort. Once a soldier was promoted to primus pilus, a silver spear was given to them. (It looked awesome, from what many classical sources say.)

Like the aforementioned soldier awards, there were also awards given to leaders as well. Not to mention triumphs, large parades through Rome to commemorate the general’s military conquest. Another is the grass crown, awarded to leaders who relieve besieged towns. Oftentimes, the aforementioned preserver crown might be awarded to soldiers during this event. If you had a leader obtain this award, it was seen as a great honor to your legion, and brought great pride to the soldiers who carried out the act of bravery.

Sources

Pat Southern, The Roman Army: A Social and Institutional History

Adrian Goldsworthy, The Complete Roman Army

Polybius, Histories 3

Caesar, Commentarii De Bello Gallico

Lee L Brice, Warfare in the Roman Republic...