I suppose that an objective definition of German ethnicity might be “a person biologically descended from one of the Teutonic tribes that spoke Germanic languages and populated Europe from the Eurasian steppes to the Rhine as of approximately A.D. 400.”
I have some reason to suspect that the OP would have some affinity for this definition. However, there are some complicating factors:
(1) This definition would encompass a much broader group of people than the OP has in mind. There are a lot of Spaniards, French, British, Scandinavian, Italian, and Slavic people who are so descended.
(2) At the same time, this definition would be too narrow to encompass all of the people that the OP would probably like to include.
(3) Language can be transmitted independently of biology. Thus, historically, a lot of people who spoke Germanic languages might not have been biologically related to each other. So, we have some trouble establishing clear identities of groups. Are they German just because they speak a Germanic language? Are people who might be genetically very closely related or identical not German just because they don’t speak German?
(4) These Germanic tribes themselves were highly intermixed with the people around them, be they Slavs or Magyars or Gauls or Latins or other groups. And since, up to the 5th century or so, they were very mobile, they were almost certainly not genetically “pure.”
The notion of ethnicity itself has always been a fluid one globally. Except for isolate groups, identification with and definitions of ethnic groups can change very quickly, even within a generation. It’s just not a category amenable to objective identification, unlike, say, language or location. Ethnicity is most often a matter of self-identification.
Look at a small place like England. Go back to the early first millennium and there are clear ethnic subdivisions: Brigantes, Belgae, Dumnonians, etc. But wait, after the Anglo-Saxon invasions, those ethnicities disappeared, and you have Britons, Angles, Saxons, Jutes, etc. Again, these distinctions soon disappeared in favor of Anglo-Saxon, Danish, Cornish, Welsh, Cumbrian. Now, what do we have in England? Anglo-Saxon, South Asian, West Indian, etc. And already, the English/Anglo-Saxon ethnicity is being subsumed into a British one, one that includes genetic contributions from all over the former Empire. And that one won’t last long before it’s just European.