Since this forum isn't intended for long-time discussions and the question thread is more or less dead, I thought about answering my own question, providing some additional information from my point of view.
TeX User Group in Austria
Some of the comments and one answer mentioned, that user groups are based on the language of a country/region. Therefore Austria wouldn't really need a separate TUG for language reasons, since the DANTE in Germany is a very ambitious group for German-speaking users. Nevertheless a local TUG could be useful in other ways, which are not language related. One of the general aims of a TUG is:
To encourage and expand the use of TeX, LaTeX, Metafont and related
Thus, local user groups could be very important for the spreading of (La)TeX within a country and in my opinion this point should not be underestimated. As I mentioned in the comments, a few teachers and I managed to introduce LaTeX to about 150+ people at my school in Lower Austria and the feedback has been phenomenal so far.
However, it turned out that there seems to be no real reason why Austria does not have a separate TUG. On the other side there is also no real reason why anybody should start one, except for the idea of spreading LaTeX more locally.
What hasn't been mentioned before is that Austria is somehow active in the LaTeX community or at least uses the program a lot. Thus, it's even more surprising that there is no separate users group.
@barbarabeeton pointed out an article from the 1990's, which was written by the Austrian Konrad Neuwirth (a student at a gymnasium in Vienna), who considered bringing TeX into schools as a programming language. His conclusion was that it wouldn't be very useful for students that have never used a computer for programming. Konrad also mentioned that TeX wouldn't fit the Austrian school system and its subjects.
Since then a lot has changed and TeX, especially LaTeX, has evolved. Based on the success of LaTeX as a typesetting program at our school, I thought about bringing it to other upper secondary schools (in Austria), although I wasn't familiar with Konrad's article at that time. The new idea is to use LaTeX as a typesetting program only, which fits the Austrian school system better than the idea of using TeX as a programming language in schools. This approach could be one reason to start a local users group in Austria, although I do not know if I want to take care of this project yet.
All I do know is that the topic could be more relevant at a later date than it is now and possibly there will be a separate TUG in Austria in a few years or more. Who knows?