LOL! Depending on whether you count from July 4, 1776 (Declaration of Independence), October 19, 1781 (British surrender at Yorktown), or June 21, 1788 (the Constitution ratified by majority of states), the United States has existed as a country for only about two and a quarter centuries.
Present-day Mexico is even younger. It covers the area where several advanced pre-Columbian Mesoamerican civilizations (Maya, Olmec, Toltec and Aztec) developed and flourished for centuries before first contact with Europeans. The native cultures were invaded and conquered by Spain starting in the early 1500's. During the colonial period from 1521 to 1821, Spain laid claim to "Nueva España" (New Spain), whose territories included today's Mexico, Florida, Louisiana, and most of what is today's southwestern United States. Mexico became a nation-state when it achieved independence from Spain in 1821.
The area that eventually became the southwest of the United States was sparsely populated with native North Americans even when claimed by Spain, which did little to actually colonize the area. In the mid-19th century, parts were sold and parts were ceded to the United States. California and Texas achieved their own independence before voluntarily joining the United States.
So what does "throughout history" actually mean? Whose claims take supremacy? True descendants of Maya, Olmec, Toltec and Aztec — or those of the many North American tribes who were native to the southern stretch from Florida to California prior to European migration? Spanish descendants, whose ancestors did little more than claim ownership of the land — or descendants of the early U.S. pioneers in the south and southwest, who developed and settled the land? How about the French, who claimed large chunks of the same territory prior to the Louisiana Purchase?
What a crock this whole "Atzlan" myth is.