Yes, both apple watches and any kind of watch can cause skin irritation. There are many variables to why this maybe happening, like the kind of metal the watch is made of or if there is sweat trapped between your skin and the watch, and even if the watch is too tight or even too loose.
After reading this article from Very Well Health, it seem that a possible reason for the watch irritation in the X post you provided is that she maybe wearing the watch too tight. With any watch but more specifically watches with sensors, it is very important to make sure that your watch is tight enough to stay in place but to also make sure that there is airflow between your skin and the sensor of the watch, as you will have a higher chance of getting skin irritation if not.