It's cute, and gets at something real, but it's overstated. The first question is "are these things mutually intelligible", with an answer on a spectrum between "yes, perfectly, obviously" and "no, not at all, obviously". There is a huge gray area that stretches pretty far from the middle in which contingency, identity, community, and nation building projects (both military and literary) move the lines around quite a bit... but as we near either pole that factual question dominates. I think. I'd be interested to learn of counterexamples.
There are German "dialects" that are completely mutually unintelligible. Take someone who speaks the standard German dialect and drop them into the Swiss Alps, and they will understand next to nothing. It will be much easier for them to learn to understand Swiss German (linguists call it "Allemannic") than it would be for someone who doesn't speak German, but it will still take time and effort to adapt. Why are they both called German?
There is Luxembourgish, which is basically the local dialect codified as one of the official languages of Luxembourg. It's otherwise perfectly understandable for people from adjacent parts of Belgium and Germany. But I guess the locals would see that it really is almost the same.
Similarly to Chinese, Germans see themselves as mostly the same culture. Standarddeutsch is pretty much a fusion between the different varieties and has evolved along with them for a long time; differently from 普通話, which is much younger and the standardized form of a northern variety of Chinese. Germans also really cling to their dialects, and Switzerland and Austria both use slightly different versions of Standarddeutsch.
The opposite example are the varieties of Serbian, Croatian, Bosnian, and Montenegrin, which are quite inter-intelligible, but which are considered as different languages by their speakers.
> Standarddeutsch is pretty much a fusion between the different varieties and has evolved along with them for a long time; differently from 普通話, which is much younger and the standardized form of a northern variety of Chinese.
Standard German is not that old. It was largely developed in the 19th Century, and it was not until the 20th Century that most people in Germany were able to speak it. Standard German is also heavily based on a regional dialect of German (in particular, central German dialects).
Standard Chinese is a product of the early-to-mid 20th Century, so about 50-100 years younger than Standard German. This just reflects the fact that German unification was in the mid-1800s, while China's modernization occurred in the early 20th Century.
The origins of Standarddeutsch trace back to Martin Luther's bible translation that exposed a huge audience to a very important text, which was most importantly also written to be approachable to that audience.
The German dialect it used was a colonial dialect that already contained mixed features from multiple dialect area and was thus suitable for a wide audience.
Of course the emerging standard underwent development since then, and low literacy rates meant that few people actually spoke and wrote it.
> The origins of Standarddeutsch trace back to Martin Luther's bible translation that exposed a huge audience to a very important text, which was most importantly also written to be approachable to that audience.
If you want to go this route, then you could also say that the origins of Standard Chinese date back hundreds of years, to the language spoken by the imperial bureaucracy, or to massively popular works written in vernacular Chinese, like the Dream of the Red Chamber (late 1700s). There are always ancient antecedents that you can trace, but they become less and less directly related to the development of the modern language.
Yes, the Luther Bible was an important influence on the development of a standard literary German, but if you want to trace the development of a standardized spoken dialect of German, you have to go to the 19th Century and the development of Bühnendeutsch ("stage German"), which because of its use in theater had to have a standard pronunciation.