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The German public doesn't care enough about language for professors to write popular books about the subject. Books about the language usually limit themselves to dealing with prescriptive questions such as: Are forms such as diesen Jahres or gewunken acceptable? The most popular German dictionary calls itself Rechtschreibwörterbuch, as if the most important thing about a word was its spelling (and not its meaning). If an expert makes a public appearance, the purpose usually is to criticise. In the instance of the Unwort des Jahres, the object of the criticism is either the concept behind a word or the way a word is used for a political purpose.
One person which comes to my mind might be Stephan Elspaß. One of his main topics is the diversity and regional differences in the languages; a topic on which he published many academic but also a few popular texts.
However this question does not have a definitive answer - comparing the achievement of people, and then even for different fields is intrinsically a subjective task.
I do not think there are any German equivalents. It is a cultural difference. In German-speaking countries, popularity is perceived as un-scientific. That is why university professors tend to avoid writing popular books. The most popular books about the German language are written by laypeople like Bastian Sick and fall far behind the quality of David Crystal’s books.
The only one who comes to mind is Peter Eisenberg. He wrote the two-volume Grundriss der deutschen Sprache, which is the most popular and arguably best linguistic introduction to the German language, the 1998 edition of the Duden-Grammatik and much of the Wahrig.
However, he is also an old white man and sometimes it shows.