Why does the River Grande have another name in Mexico?

Answer 1

After the Mexican-American War (1846-1848) all territories north of such river became part of the United States. Before the war there were no attempts to fully map the course of the river, so many people named it in different ways over the centuries.

The upper part of the river, from present-day Rio Grande National Forest in Colorado down to the Conchos River in Mexico was named as Rio Grande del Norte by Spanish explorer Juan de Oñate in 1598. The lower part of the river, from the Conchos River in the Mexican state of Chihuahua down to its delta in the Gulf of Mexico was known as the Rio Bravo del Norte.

So, as a matter of fact, Mexicans use the name given to their part of the river, while Americans use the name given to the upper part of the river when referring to the whole river.