Fog itself is not dangerous to breathe in because the fog is basically just water vapor in the air.
As long as there are no other pollutants in the air mixed in with the fog then it's safe to breathe.
However in some places such as huge populated cities where there are a lot of pollutants in the air that create smog which is fog mixed with pollutants then that smog becomes dangerous to breathe and can cause respiratory issues and even possible lung cancer or COPD.
But for plain old fog it's safe to breathe in.
But some people get breathing issues when it's foggy out but it's not because the fog is really harmful it just messes with their lungs.
When the fog goes away then their breathing issues usually go away as well.
Fog is mostly dangerous when driving as it reduces your visibility.