For what it's worth, ethnicity and statehood/citizenship was generally always only loosely linked until the rise of the idea of the nation state in Europe, a process which began roughly in the 15th/16th century and becomes a colloquially synonymous term generally only after (i.e. a person from an area is personally of the characteristics of that area not only in terms of ethnicity but also of that areas political status and rights). In Ancient Rome one could thoroughly be born and bred in Rome but as a slave descendant one would not be considered a "full Roman": a Civis Romanus, a Roman citizen, had particular rights ascribed a slave certainly had not.

Thus, citizenship and ethnic identity were previously somewhat tied to one's social status as well. Similarly, in the middle ages, one's (tribal) ethnicity similarly did not equate to a full sense of belonging to a particular political entity, as feudal loyalty to particular dynasties did not necessarily equate loyalty to their ethnicity: your feudal lord could very well call upon your loyalty even if he was not of your particular ethnic group (such as Norman kings calling upon their Saxon followers, etc)

Only in the age of nationalism does ethnic identity openly and ideologically drive attempts to perfect a match between political constitution of a geographic area to a particular ethnic belonging. Each area (nation) of Europe has a vastly different history here, but similar processes happen throughout Europe in the early modern period as ethnic boundaries that do not match their political boundaries sometimes get "reinvented" to fit, or vice versa, and histories are "projected" from the present into the past (the Portuguese "invent" themselves as descendants of the ancient Lusitanians, the Swiss claim to be (Celtic) Helvetians, the varying populations of the Frankian lands get forged into "Frenchmen", English, Scots and Irish into "British", and the many Central European Germanic tribal ancestries get lumped into a "German identity" that of all options chose the Teutons (= "teut'sch" = "teutonisch", the root for the term "Deutsch") for their claim to nationhood. These processes were not without trouble and often never completed (cf. Irish identity and "Britain"), or certain regions resisted such ideological drives against their own budding ethnic nationalisms (Cf Basques and Catalonians in Spain, etc).

I don't think an 8th century Saxon or Eastern Frank would have liked to be called a Teuton, especially as the Franks under Charlemagne slaughtered the Saxons quite a bit around that time. But the need of nationalism to claim a joint heritage to call upon was predominant throughout Europe by the 17th century (even if entirely inaccurate), and in many ways gave rise to the actual discipline of modern history at state funded academic institutions -- which ironically is today fundamentally responsible for revealing the fallacies in the very same claims it preached in its infancy. Modern historical academia in many ways was state sanctioned to legitimize the historical claims of the nation state, and in the age of enlightenment was called upon to find "reasons" why nations could legitimately claim certain boundaries and setups vis a vis the "others". But this very enlightenment also contained the seeds of its own "debugging" of its impure unreasonable wishful thinking.

The problem with trying to match "German" ethnic identity with its geographical boundaries in particular was that it simply would not fit especially with regards to the lands to the East and Southeast. It was easier on the Western frontier, where a rough boundary to the former Roman Empire could match roughly along the separation lines to the Western franks; but in the east, German expansion had driven deep into Slavic tribal lands as late as the 15th century (obviously centuries after the actual "Teutons"), and the quilt of settlements and fluctuating rulership there made a clear boundary of "Germanness" to the east impossible for purposes of nationalist ethnic identity. On top of that, Germany came late into the game of political nation-forging, as the Holy Roman Empire and even Prussia had been decidedly non-ethnic states. (A famous quote of a Slavic-descendent nobleman of Eastern Prussia after the founding of modern "Germany" in 1871 goes: "We were always good Prussians, but we can never be good Germans." -- ethnic identity suddenly trumped political loyalty and equated the two, thus excluding some former loyal citizens suddenly from being "German"), and worse, claimed heritage of all things as a discerning factor of one's belonging.

This (what one could call an "unholy") marriage of citizenship with ethnic heritage can indeed be blamed as an ideological cause leading to the thinking behind many atrocities in the late modern period after 1800. So in short, yes, the notion of "Who is a German?" kept Germans quite busy, and as current events show, it is not the only area in Europe (or the world) that is suffering from this particular predicament.

[Edit: u/Schnurzelburz corrected me, correctly, on the root of "Deutsch" from lower German "theodisc" -- an important bit, of which I am happy to have learned today]