English isn't the only language the Japanese think is cool. Much of their western vocabulary was adopted from Europe, so not all pronunciation differences stem from Engrish. Quite a few anime series include German words and phrases instead of, or in addition to, English words to add an exotic flair. Like English, the quality of German in an anime series can be quite good or very bad.
In many cases, German sounds angry to most English listeners (and plain annoying/stupid to others), regardless of what is being said; this is rather different from how it is spoken in Germany, where it is more nuanced and even quite pleasant to listen to. Ja! That movie of Hitler getting angry and speaking in German might have something to do with it. The fact that he's speaking with an Austrian accent never dawned upon us, and neither did the fact that Hitler's speaking style is quite unique and hard to comprehend for the average German OR Austrian to begin with.note
In Japan, the opposite effect happens. Since the more guttural parts of German cannot be written or spoken in their language, they just disappear. The result sounds ridiculously sleek and stylish by their standards, though bizarre and butchered by many others'. Even common names like Schneider come off so majestic and bishounen-exclusive, it's a joke when they're not. This difference in perception results in German practically fueling Rule of Cool there rather than being Played for Laughs. The same thing happens to other languages, like Hebrew losing the throaty parts, contributing to their angels becoming excessively pretty even by Japan's standards. note
The popularity of German is possibly also related to good relations between Japan and Germany during the 20th century. Or because when Japan was modernizing in the late 19th century, it borrowed heavily from Germany, which was also a newly forming nation-state at the time, and world class in technology and science (e.g., the Japanese legislature is called the Diet in some countries as it was partially patterned on the 19th-century Prussian Diet. In Japanese, it is apparently called the "Kokkai", literal translation is National Assembly which makes a lot more sense.).
A variation on this trope is that when an anime character is mixed Japanese and something else, the "something else" often turns out to be German. This may result in the character speaking some garbled German at some point.
In Western media, Gratuitous German is often associated with Those Wacky Nazis, but also crops up in philosophy and psychoanalysis - any substitute for Sigmund Freud will litter his flavour of Poirot Speak with it. In other words, Everything Sounds Smarter in German, so don't be surprised if he starts talking about how the Kantian Ding an sich ist inherently incompatible with the Weltanschauung of the Übermensch because for him, Gott ist tot, and the Wille zur Macht ist such that.....
Although both are West Germanic languages and thus closely related, the German Language has kept a bit more of the original comparatively high amount of grammatical inflection than English. This often makes it difficult for foreign language speakers, resulting in frequent and very visible grammatical errors even if the sentence is understandable in general. Never expect the grammar to be proper; expect a "Blind Idiot" Translation. Another common pitfall is the pronunciation, that contains the Umlaut (German plural Umlaute) "ä"note "ö"note "ü"note , which are actually sounds that exist in several dialects of English for the most part and "ch" (kind of like Spanish "j") which only exists in Scots. Works that get those right have really Shown Their Work and most likely had a native speaker look at the German at some point.