I may not be familiar with the name taco but from little researched i made it seems to be called shawarma to my local understanding because that is exactly how shawarma looks like it is comprises of all details i find in my research.
Tacos are indeed very similar to Shawarma, but the spices, seasonings, and toppings are slightly different. In Greece they call shawarma "gyros," but they are essentially the same thing. The way the meat is seasoned can actually be very similar also. The primary spice is cumin in both, but in varying quantities with other more local seasonings added. Tacos and Shawarma can be made with chicken, but originally shawarma are made with a mixture of lamb and beef. The toppings can also be similar; everything from fresh tomatoes to cabbage, fresh or pickled peppers, cilantro, and fresh onions. I've never seen cucumbers on a taco where they're quite common on shawarma. The main difference is the sauces; Mexicans use a tomato or tomatillo based salsa, Arabs use tahini (sesame butter) based sauce or a yogurt sauce, and Greeks almost exclusively use a yogurt sauce.
The best shawarma I've had was in Ramallah, Palestine, where the locals claim it was invented. I've also had them in Nablus and Al Quds (Jerusalem,) all amazingly good. In Amaan, Jordan they can be found on almost every street corner, and also among the best I've had. Once I made the mistake of trying one at Reading Terminal Market in Philadelphia, never again! The worst was from a street vendor in New York where they put mayonnaise on it. Fucking mayonnaise! There are a couple of local mom and pop delis here in northern California that do a decent job.
Shawarma meat is cooked on spit like the photo below. Mexicans use the same method for cooking adobado, al pastor, and trumpo which are all essentially the same thing (marinated pork,) but come from different regions of Mexico. There's a sense of regional pride and friendly competition about which is the best. I don't eat pork, so I can't tell you if they are any different from one another.
Chicken and beef for tacos is usually cooked on grill, but one of the unique things about tacos is the variety of meats one can choose. Carne asada (beef) and pollo asada (chicken) are grilled or sauteed, lingua (beef tongue,) cabeza (braised beef head,) pollo verde (chicken braised in tomatillo salsa,) buche (braised pork,) carnitas (deep fried pork,) or adobado variants as mentioned above.
White people tacos are usually made with ground beef, which is an inexcusable tragedy. My favorite tacos are carne asada and cabeza.