But this raises the important question whether in a functioning democracy it is ever okay for the police to investigate who marked a specific ballot.

Does Germany have a post-war law that makes this a crime? I thought the had a lot of anti-Nazi laws on the books.

Edit: Turns out it is. Under Section 86a of the German Criminal Code (StGB), it “prohibits the distribution and public use of symbols of unconstitutional organizations and those similar to them. This includes symbols of parties or organizations banned in Germany, especially those associated with National Socialism, as well as their propaganda materials. Violations can lead to imprisonment or fines.”