traditional German sans-serif (and serif) blackboard-bold symbols

I suggest you take a look at Michael Sharpe's mathalfa (the spelling "mathalpha" works too) package. In the package's user guide, the author lists five "hollowed-out" and fourteen [!] "geometric" variants of blackboard-bold fonts. (Note that not all of the underlying fonts are free.) I trust that one or more of the math blackboard board fonts listed and/or made available through the mathalfa package will meet your needs. (I'm afraid I don't have access to German-language math textbooks.)

A big plus of using the mathalfa package is that you can "mix and match" various math alphabets (blackboard bold, caligraphic, curly, script, fraktur, etc) with ease. Excerpting from the README file that comes with the mathalfa package:

This package provides a standard means of setting math alphabets associated with the macros \mathcal, \mathbb, \mathfrak and \mathscr and, where available, their bold counterparts \mathbcal, \mathbbb, \mathbfrak and \mathbscr.

It mostly bypasses the usual fd and sty files used to load these alphabets in order to allow each to be scaled independently and without silently quantizing the sizes.

The definitive answer turns out to be quite distributed (found through collaborative discussion). Credit goes to multiple people (see the entire discussion). Awarding bounties would have been infeasible in this case as they would have to be distributed over a few people.

Sans-serif: Package mathds with option sans (\usepackage[sans]{dsfont}), except for doublestroke-"A" one needs to use \mathds{a} instead of \mathds{A} to get the doubled stroke on the "correct" (as per expectation) side (i.e., on the left).

Serif: The following fonts mentioned in mathalfa's documentation would "pass" to the eye of a German user used to the traditional lettershapes of German blackboard bold: pazo (Mathpazo bb), tx (tx bb), px (px bb). "tx bb" looks closest to my memories, though I like the "S" of Mathpazo bb as being more consistent with the implicit design. The mathds package also yields letters whose shapes "pass" for "traditional German" appearance, except for the doublestroke-"A", one needs to remember to use the lowercase option (\mathds{aBCDEFGHIJKLMNOPQRSTUVWXYZ}), just like I mentioned above in the sans-serif paragraph.

(Note: I overlooked, that the OP already wrote about this package. But according to the comments my answer seems still to be useful.)

There is also the package doublestroke (Texlive)/dstroke (MiKTeX), which provides a serif font (and a sans-serif version), that

is useful for typesetting the mathematical symbols for the natural numbers …, whole numbers …, rational numbers …, real numbers …, complex numbers …, and a couple of others which are sometimes needed.

To use it, one has despite the package name to add first \usepackage{dsfont} and later a \mathds{<symbol>}. There are glyphs for all capital letters A…Z, for 1, h, k and for the letter a (i.e. the small letter!). The latter gives a different A with the doubled line in the upstroke.

_{(The source for the following example is the manual.)}

\documentclass{article}
\usepackage{dsfont}
\begin{document}
\[\mathds{ABCDEFGHIJKLMNOPQRSTUVWXYZ}\]
\[\mathds{a}\]
\[\mathds{1}\;\mathds{h}\;\mathds{k}\]
\end{document}

To get the sans-serif version, one has to set the package option ‘sans’:

\documentclass{article}
\usepackage[sans]{dsfont}
\begin{document}
\[\mathds{ABCDEFGHIJKLMNOPQRSTUVWXYZ}\]
\[\mathds{a}\]
\[\mathds{1}\;\mathds{h}\;\mathds{k}\]
\end{document}

_{BTW: Found by searching with Detexify LaTeX handwritten symbol recognition and then looking for dsfont on my computer.}