It really doesn't. If anything German is a fairly soft, camp-sounding language. The whole shouting thing seems to be based on World War II stereotypes.
There is a characteristic gentle sound when many German people speak English as well. They also tend to enunciate more carefully than many native English speakers, so they are usually very easy to understand.
I find that same careful enunciation means German spoken by natives often has a slightly sharper sound than English, but that could just be because many everyday German words use sounds like ch, ck, b, d and z, and as a native English speaker and non-native German speaker I notice those more. As you say, this has nothing to do with shouting, though.
>It really doesn't. If anything German is a fairly soft, camp-sounding language
Not for us non-Germans listening to it (though the video exaggerates of course). There's a reason behind the stereotype, and that's not WWII.
Even the mere written form of the words is more somber and threatening looking: schmetterling vs butterfly or farfala for example, or Wissenschaft vs science.
Can't compare with something like Italian or French.