Company names were strictly individual in nature. Here's a well-known story about a famous Alabama company commanded by Capt. John Brown Gordon --
Gordon's company was among the earliest of these volunteers....Not waiting for orders, they bade a quick adieu to all they had held dear and set off cheerily for Milledgeville, then the capital of Georgia. They were destined to a sad disappointment. On reaching Atlanta they were met by a telegram from the governor, who had been advised of their coming, telling them to go back home and wait until advised that they were wanted.
This was like a shower of cold water poured on the ardor of the volunteers. Go home? After they had cut loose from their homes and started for the war? They would do nothing of the kind; they were on foot to fight and would not consent to be turned back by Governor Brown or any one else. The captain felt very much like his men...but his position was one demanding obedience to the constituted authorities, and by dint of much persuasion and a cautious exercise of his new authority he induced his men to board the train heading back for their homes.
The repressed anger of the rebellious mountaineers broke forth again when...the whistle gave its shrill starting signal. Some of the men rushed forward and tore out the coupling, of the foremost car, and the engine was left in condition to make its journey alone. While the trainmen looked on in astonishment the mountaineers sprang from the train, gathered round their captain, and told him that they had made up their minds on the matter and were not going back. They had enlisted for the war and intended to go to it; if Governor Brown would not take them, some other governor would.
There was nothing left for the young captain but to lead his undisciplined and rebellious company through Atlanta in search of a suitable camping-place. Their disregard of discipline did not trouble him greatly, for in his heart he sympathized with them, and he knew well that in their rude earnestness was the stuff of which good soldiers are made.
Atlanta's streets...were filled with citizens, who looked upon the motley crew with a feeling in which approval was tempered by mirth. The spectacle of the march — or rather the straggle — of the mountaineers was one not soon to be forgotten. Utterly untrained in marching, they walked at will, no two keeping step, while no two were dressed alike. There were almost as many different hues and cuts in their raiment as there were men in their ranks. The nearest approach to a uniform was in their rough fur caps made of raccoon skins, and with the streaked and bushy tail of the raccoon hanging down behind.
The amusement of the people was mingled with curiosity. "Are you the captain of this company?" some of them asked Gordon, who was rather proud of his men and saw nothing of the grotesque in their appearance.
"I am, sir," he replied, in a satisfied tone.
"What company is it, captain?"
As yet the company had no name other than one which he had chosen as fine sounding and suitable, but had not yet mentioned to the men.
"This company is the Mountain Rifles," said the captain, proudly.
His pride was destined to a fall. From a tall mountaineer in the ranks came, in words not intended for his ears, but plainly audible, the disconcerting words,
"Mountain hell! We are no Mountain Rifles. We are the Raccoon Roughs!"
And Raccoon Roughs they continued through all the war, Gordon's fine-spun name being never heard of again.