Though I personally think that yes, correlation does imply causation. It just doesn't prove it. But the more correlation you have, the more likely there's some causal relationship. If you see A and B together all the time, maybe A causes B, or maybe B causes A, or maybe something causes both of them.
Temporally, the adoption of a vegetarian diet, on average, tended to follow the mental health diagnosis, suggesting that the vegetarian diet was not in fact causal. I know originally the abstract of the article said the opposite, but if you read the full text, you find that the abstract was misrepresentative.… So what is going on? In Germany, are the neurotic perfectionists who are more likely to be choosey about food (and thus sellect vegetarianism and eschew fast food) also more vulnerable to Depression and anxiety? Sure, could be. Or maybe those with mental troubles try to avoid what is thought to be bad food (meat and fast food). It is also possible that the nutrient deficiencies common in vegetarian diets (the most robustly studied being long chain omega 3 fatty acids and B12, though I think zinc and creatine and even too low a cholesterol could also be issues) could accelerate or worsen pre-existing mental conditions.