Multiple cultural influences and cuisines here, too (Houston, Texas). Have never seen anything like that involving pork - thanks for identifying the meat - I was not sure).
Overwhelmingly in the US, Americanized Greek restaurants serve 'gyros' made with a mixture of beef and lamb, minced, mixed with spices and pressed into a cone shape. There is no layering of individual pieces of meat, as mentioned above by CDH, I think. I have read pork is most common in Greece itself although other meats do appear. That would not commonly be thought of as gyro meat in America.
If that's pork then that's not shawarma!
I can only think of 2 dishes I've had at Chinese restaurants which I thought might have been shaved - at a Uyghur place and a Lamen place (oil splashed beef lamen). They're both closed. I have no idea if either one had an upright roaster like that.
Ya got me - I give up.
