First, about the names, you can consider them a practical way to differentiate people in a huge family where a lot of them shared the same names. They probably worked in the same way and were influenced by the Roman cognomen. During the Roman era your name was a three part mix of a generic praenomen, taken from a small pool of 12-15 of such names (Like Gaius or Lucio), your nomen (Julius – Cornelius), which was the family name, and a cognomen (Ceasar - Sulla), which was usually linked to a personal trait but, in esence, it was just a nickname given to you to differentiate between you and your cousins, brothers, and other family members. Same case scenario will roughly apply in Carolingian times. You had a bunch of people which were of the same family and who shared the same pool of names, with Charles and Loius been the most common, so they needed the nicknames to be able to know about whom they were talking about. Also I don´t know exactly to which extent this practice was just followed by the Carolingians or more widely used in the whole Christian world, although from what I recall this naming convention was also used by most societies of Germanic descent, like Franks, Burgundians or Saxons.

However, one possibility may be that the nicknames became so common as a way to reference the Carolingians as consequence of Academia and how historians used them to clarify whom they were writing about. Even if the nicknames were invented and used by the Carolingians themselves, they may not been used so often as we do now to discuss them. Most of the Carolingian sources I read don´t use them, Louis the Pious is just called Louis, or the Emperor, and his sons Charles the Bald and Louis the German, are just Charles and his brother Louis. Here´s an extract of the English translation made by Janet Nelson of the Annals of St. Bertin written by Archbishop Hincmar of Reims from the year 871.

"King Charles at the request of his nephews, his brother Louis´s sons Louis, and Charles, travelled by way of Verdun to meet and talk with them. Then he returned of the synod at Douzy [...] There King Charles gave an audience to envoys sent from his own son Carloman, with his brother Lois acting as go-between"

Hincmar does not use the nicknames, he just traces links of parenthood as a way to specify of whom he is talking about. Although the nicknames are of Carolingian origin, I doubt they were used as much as we use them today. Even less in written texts and chronicles; which are of course our main source of information. I find more likely that the nicknames became widely used because of later historians, who used them as a way to try to clarify the complex mess of same-named people that the Carolingian family is, especially for students or anyone who is new on the subject. That way, the first time that most of us heard about King Charles of West Francia, we heard of Charles the Bald or about his brother Louis the German, as way to differentiate between them and the later sons Louis the Younger or Charles the Fat (who at some point was also King of West Francia and Emperor). A different question would be to say to which extent they were used in an oral and more informal context between members of the family which, due to the lack of evidence, is something incredibly difficult to ponder about. There is probably some books or articles about this somewhere but I don´t remember anything specific that I could cite now.

Now, about your second question, how did ninth century people regard ethnicity, race or "nationality" so to speak. Here I can cite an interesting article of Helmut Reimitz "Omnes franci: Identifications and identities of the early medieval franks" included in Garipzanov, (ed.), Franks, Northmen and Slavs identities and state formation in early medieval Europe (2008). His main thesis is that identity was something much more fluid and interchangeable that it is today. Being a Saxon or a Frank was more a question of context that of personality, and the same person could assume different identities depending on the circumstances. A noble from Saxony or Bavaria will probably feel Bavarian or Saxon when he was with his peers or locally, as they had their own, laws, customs and linguistic variances, but he´ll probably identify as a Frank in a royal assembly held in Frankfurt or anywhere in the kingdom. This would be especially true in the early stages of the empire, during the reigns of Charlemagne and his son Louis, where the elite of the whole empire was quite intertwined. Being a Frank would just mean being a member of the elite who ruled the empire, something which was no exclusive with being Saxon or Bavarian locally, due to your origin, family or because of the lands you ruled. This tendency probably stopped working as strongly as the instability in the empire grew, and each kingdom became more isolated from each other. Although I wouldn´t say this mattered too much before the reign of Charles the Fat, after which the disgregation of the kingdoms finally was well on its way. The way Helmut refers to identity as something less rigid than it is nowadays I think it´s reasonable and makes quite a compelling argument, but topics such as identity, race, or religious belief are always extremely difficult to tackle with the information we have, so we should not take any account as an absolute truth as we can never be completely sure about them.

And lastly, going back to Louis the German, personally, I don´t think his nickname is that significant in comparison with the other Carolingians. He ruled the East part of the Frankish kingdom for nearly half a century, and he was also raised there and made sub-king during his childhood. He is one of most long-lived Carolingians so I don´t think it is odd that he was called the German, as he literally was the king of the region during a good portion of the century.