Appeal the grade in oral exam in Germany
Here's my experience with oral exams in Germany:

Ask the professor to explain to you how they arrived at the grade, and what they'd expected you to say for a better grade.
Maybe more important about this than the grade you got now is what you can learn for future oral exams.
(When I asked the &quot;lemon squeezing professor&quot; below, he explained that from the grades I got in the labwork exam he was examining &quot;for a top mark&quot;, and would not waste time asking me easy questions about the lecture but that he examined my ability to draw field-specific conclusions on the basis but outside what was explicitly discussed in the lecture... wow...)




The burocratic trail corresponds to the importance of the exam. Oral exams with only 1 student + 1 examiner and no paper trail typically end up contributing low percentages to the final grade for the semester and even lower percentages to your Bachelor/Master grades. The next more important ones already had a 2nd observer (usually PhD student or postdoc) though still without protocol.
Everything of higher importance (including legally relevant exams) is done with someone writing a protocol, and in some cases even audio records are taken. This is done exactly in case there is dispute about the grading.

Rather than complaining about the grade in order to get a better grade, I've only met students who asked to have their grade lowered to &quot;failed&quot; so they could re-take the exam.

For the &quot;low importance&quot; oral exams I've been involved with, a complaint would usually have resulted in the offer to have another exam which for sure would have someone writing a protocol.

Back when I was studying, for the low-importance exams in many cases it would have been possible to retake them the next semester/year (sometimes even the high-importance ones, under certain conditions). However, I've never met anyone who did care sufficiently to do so.

As always, the exam regulations of your department are what matters. You need to look them up and/or ask at the exam office.

Grades are submitted to the exam office, and the professor thus cannot easily change the grade. Thus, the risk of getting your grade lowered if you complain to the professor is small. But so is your chance to get it raised: both will likely need a (very) formal procedure with you, the professor and the exam office.

A professor &quot;can give any grades they want[ed]&quot; only in the sense any public servant in government administration can do anything they want to an application/procedure you filed there: professors in Germany are public servants  and exams are one of their public servant duties. Violations of these duties are considered quite serious in general (they took an oath to fulfil these duties).
(Which does not mean that unfair grading does not happen - but it does mean that if you file a complaint about unfair treatment in an exam, that is a serious accusation)




One thing you may consider for future exams: as students, we were allowed to &quot;bring a friend&quot;, who would then automatically be a potential witness. If there's a problem with the professor's grading, such a potential witness may make the whole procedure more fair already by just being there.
This happened very rarely, but I remember a professor recommending this for an entirely different reason: so that the friend can find out what kind of exam to expect. I.e., they recommended that we look for someone who'd be comfortable taking us along before doing the exam ourselves.


A last thing is to bear in mind that the perceived difficulty of question and your difficulty of answers may not predict well even a rather objectively justified grade. There are (legitimate) differences in the style how different professors approach the task of examining. Some will ask everyone questions on a similar level of difficulty and then consider the answers, others adjust the difficulty of the question according to preliminary grades and previous answers (I found it extremely hard to judge as a student how such an exam went - but I do know since how a lemon must feel after sqeezing. Got 1,3 btw.)