First, I attached the lodestar to the main scope. There the image was perfect. So, it seems as if it is the guidescope. I moved it back, and ... no elongated stars at all! Strange. So, I focused the scope as good as possible (with the small Bahtinov mask).
Well, I then tried to work on autoguiding with TheSkyX. At some point, I had the elongated stars again. It took me a while to figure out, that this only happens when I'm 4x4 binning. Here are some images:
Well, after that was cleared up, I tried to get autoguiding to work. But still no luck. I posted the guiding log file, the lodestar image and the autoguiding image to the Software Bisque forum - hopefully I will get some help.
And I also realized that I need to build a shield for the guidescope. Here is an image where some light reflects off the main scope:
And here is the same image with all lights turned off and some black paper on the scope:
And that was it.
Few days later: Yes, this is because of binning. Apparently, the lodestar only supports 2x2 binning. And if you choose a higher binning value, it only bins in one direction, i.e. 3x2 and 4x2 - that explains why everything looks so elongated. I can actually set the max binning in the ASCOM driver configuration. After I did that, TheSkyX only had 1x1 and 2x2 binning in the dropdown. Riddle solved!