As you expect in asking this question, there is much more to the story. There is some stress reduction from having a companion of any kind to talk with as it constitutes a distraction. But this, in and of itself, is no more profound than having television to while away the hours and keep one from focusing on their own problems and troubles relentlessly. There are some health benefits through motivating people to get more exercise, so in having to walk the dog, this confers some health benefit, but that is modest at best.
In most cases, the primary reason for the statistically significant health benefit with regard to cardiovascular outcomes can be attributed to the healing work done by these canines for their human companions. The fact that heart disease shows this, is not that it is the only health need being addressed by these healer companions, it is that heart disease is a very prominent health problem that can be rapidly fatal. So it is easy to track and all that is required is a prior diagnosis and a death certificate to register data that is easy to compile and analyze.
Many, many chronic conditions go on for years and years and may have waxing and waning episodes and tempo. Those are more difficult to evaluate and obtain quantitative data showing the influence of an intervention, no matter how profound it might be. It is easier to see a significant change when a fatal condition is reduced in severity sufficient to preserve life even though the underlying pathology may still be present and there may, in fact, be other symptoms as well. It is nonetheless a tremendous benefit to avoid early death, but this is the tip of an iceberg going on because broad-based healing is a standard part of the repertoire for companion animals who take this on freely through their own initiative and do what they can to help their owners.