Reasoning about cause and effect when it comes to this kind of thing is really hard, and frankly, a lot of people in epidemiology and especially journalists just aren't that good at it. Researchers get fooled by reverse causation and confounders all the time, and it's very easy for fallacious conclusions and spurious findings to get laundered into "fact" through repeated citation. This is especially likely to happen with feel-good stuff like "pets prevent Alzheimer's" or "wine prevents heart disease."

Maybe dog ownership improves health by promoting exercise, but there's very unlikely to be any real direct, clinically important biological effect of warm fuzzy feelings or reduced loneliness or whatever.