Is there a standard format of how stereo calibration data (various matrices, usually saved in XML) are stored? Can I load calibration data generated say from a OpenCV script in C to another OpenCV script say in C++ or to completely different software where I create disparity image?
The short answer is no.
You can use the XML output file within OpenCV's ecosystem (and ROS), but there are no standard formats for calibration.
The issue is not coming up with a standard. Camera intrinsic calibration models differ in their modeling of lens distortion, due to different lenses and different application settings.
If you are using a calibration obtained from OpenCV, I'd highly recommend making sure you read calibration documentations before plugging in the values into another package.