The short answer is yes, but only if the operator is inexperienced.
Modern paints are environmentally friendly without solvents or heavy metals.
The clear coat, top or finish coat is the part of the paint that is in contact with the outside world. This layer is an acrylic clear paint. We all know that acrylic is a soft paint that doesn’t wear well in the open environment. It oxidises readily if its not cleaned on a regular basis. This oxidise paint or dead paint is chalky and brittle and scratches or crumbles easily under slight pressure.
How many times have you seen someone use their finger nail to remove a bit of dirt stuck to the car body after its been washed, it’s instinctive, people do it all the time, resulting in a scratch.
The problem is the expectation that the wash will remove all dirt from the surface of the car, this is a wrong assumption, and attempt to remove all types of dirt will result in damage to the car. An extreme example for instance is cement or concrete on the car this is actually very common. If you try to remove cement in any other way except by the correct way you will end up scratching your car. Now there are many different types of dirt that will harden and will not come of in the wash just like concrete.
These foreign bodies can be removed but you have to use the right method. There are as many different removal methods as there are contaminates.
On other hand almost every contaminant will come of in the wash if you wash it soon enough. Think about concrete again if you wash it straight away, even just water will remove it, If you leave it a couple of hours then its totally different storey.
The moral of the storey is that cars are meant to be washed at least weekly if you expect it remove 99% of the dirt, this has the added advantage in that it stops oxidation of the paint.
So the cause of scratches at car washes is the infrequent washing and the expectation that the once a month wash will remove all the dirt.
It can be done but only by force, which ends up scratching, or by strong chemicals that slowly etch your paint. Many car washes use the strong chemicals and save themselves the headache of trying to educate the customer. Slowly after half a dozen washes your paint is fading ever so slightly with every wash.
Be cautious of cheap car washes that leave your car very clean, in all likelihood more than the dirt has been removed.