When it was the Gulf of Mexico, it was named from our point of view. It was the gulf, among our many gulfs, that we share with Mexico. Now, by the same logic it is named from their point of view.
All those in America (South, North, Central) are called Americans.
I spent several years in Paraguay where I met a lot of South Americans.
Mexicans are North Americans and the 'Gulf of America' is more inclusive considering 1/2 of the land mass touching it belongs to the US.
Actually it was named from the perspective of Spanish explorers. Originally Gulf of New Spain but by 1569 had become Gulf of Mexico.