Most Americans would be surprised that the German interpretation of “the untouchable dignity of man” prohibits even insults (“Beleidigung”). There was a case in 2018 of an American who got in a tussle with German airport police, called them Nazis, and was subsequently charged with defamation.[1] Clearly she wasn’t literally describing them as students of the philosophy of Adolf Hitler, but rather calling them authoritarians in a demeaning and insulting way. In the U.S. one might expect her to be charged perhaps with disorderly conduct or something similar, but punishing her for her words would require that there be some actual injury done to the officers’ reputation. Obviously ridiculous statements that no reasonable person would believe don’t fall under defamation in the U.S. A famous example was in the 1980s when Jerry Falwell unsuccessfully sued Hustler for publishing a fake interview with him claiming he drunkenly lost his virginity to his mother in an outhouse.
Certain depictions of violence are also considered to inherently violate human dignity as a whole (“Gewaltdarstellung”). That seems to be why there’s traditionally been so much video game censorship in Germany, like green blood or replacing all characters with robots[2].
Most Germans, on the other hand, would be surprised that insults are legal in many countries. Wikipedia has a map of this.[1]
There is another aspect regarding the legal situation in German that might be relevant to understand the difference to the US: The US constitution speeks only of the "freedom of speech", while the German consitution (Grundgesetz) goes much more into the detail:
(1) Every person shall have the right freely to express and disseminate his opinions in speech, writing and pictures and to inform himself without hindrance from generally accessible sources. Freedom of the press and freedom of reporting by means of broadcasts and films shall be guaranteed. There shall be no censorship.
(2) These rights shall find their limits in the provisions of general laws, in provisions for the protection of young persons and in the right to personal honour.
(3) Arts and sciences, research and teaching shall be free. The freedom of teaching shall not release any person from allegiance to the constitution.[2]
So the right to personal honour is directly acknowledged.
As an aside: "teaching" ("Lehre") refers to higher education only, but includes both, teaching in the narrower sense and learning. As a consequence a student at a German university has the right to learn as he or she seems fit and can only be forced to attend a course, if his or her physical presence is really necessary as there is no other way to learn a particular skill. So the presence at laboratory courses or courses focused on discussions can be made mandatory, but not at lectures. (The student must only attent the course exam, if there is one.) Isn't it interesting that a student in Germany has a constitutional right to be an autodidact?
Given how freely some people use “nazi” as an insult, I am not surprised that visitors to Germany may be unaware quite how strong an insult it is here.
I’m still a mere Ausländer here myself, so I only have a rough sense of it, but can say that every few hundred meters along the streets you can find a gold-coloured paving stone memorialising separate victims, either an individual or a family.
I've also heard from Germans that they have been called Nazis in the U.S. just for being German. So the police officers could also perceive it as an ethnic slur.
Certain depictions of violence are also considered to inherently violate human dignity as a whole (“Gewaltdarstellung”). That seems to be why there’s traditionally been so much video game censorship in Germany, like green blood or replacing all characters with robots[2].
[1] https://www.presseportal.de/blaulicht/pm/74262/3844730
[2] https://www.youtube.com/watch?v=aHIpHOVB9ew