According to the STA notes when this tartan was first woven in the 1930s it was called "Green Stuart" but quickly took on the name Princess Mary. It was popular at the time to name new tartan designs after members of the Royal Family.

I wouldn't consider this a "Royal Tartan" in the same vein as Balmoral, however. If you wanted to wear it, there should not be a problem. If you feel a little odd calling it "Princess Mary" you can always call is "Green Stuart" instead. :-)