The US has more "agressive" ads than Germany, because in Germany exists law against such agressiveness. For instance, you are not allowed to put down a competitor, not even mention their products.
> For instance, you are not allowed to [...] even mention [a competitor's] products.
This is directly refuted by the section of legal code that you linked. The phrasing in there (translated in German) is along the lines of "Mentioning a competitor's product is unfair advertising if [specific catalog of unfair situations]". The list includes things like "comparing products intended for different purposes and functions" or "comparing products by qualities that are not objectively measurable".
I'm not saying that you will not be walking on thin ice if you engage in comparative advertising. In fact, if you target a particularly litigous competitor, you may have a bad time even if you're within your rights, but to say that comparative advertising is flat out illegal is not backed up by the phrasing of the law as presented in your source.
"Law against unfair competition"
Gesetz gegen den unlauteren Wettbewerb (UWG)
https://www.gesetze-im-internet.de/uwg_2004/__6.html