The answer to the question of whether or not a clogged shower drain can cause a leak is that I guess it depends. Generally speaking a clogged drain won’t cause a leak. However, I can imagine times when there are secondary causes that are associated with the backup of water.
Regarding a shower drain, as the water backs up back into the shower pan and begins to rise, if it gets high enough, it may creep into back into the wall if there are voids in the caulking or grout.
The backup of water will likely put extra stress on the joints as well. Not that the joints won’t hold up, but they are not normally under pressure. It’s similar to when you plunge a drain occasionally a joint might be disturbed to the point that it leaks, but it isn’t common enough to worry about.
Toilets and sinks are the same way. If a toilet backs up, it can certainly overflow on to the floor and cause problems downstairs. For a bath sink, there is an overflow drain that should take the water to the drain, but if the drain is plugged it can overflow. I have also seen the overflow channel below the sink leak as many of them are plastic and are not secured very well. So generally I would say a clog won’t cause a leak.