Who owns the actual boundary Rio Grande River, the United States, Mexico, or both?

I don't think it's quite that simple:

The Rio Grande (known as the Rio Bravo in Mexico) courses through three U.S. states and two nations and has profound historic, cultural, and environmental significance for the citizens of its basin. It has been dammed, diverted, and polluted in the last century, reducing one of this continent’s great rivers to a vestige of what it was when the Spanish conquerors first sighted it. Occasionally it doesn’t even contain water, raising questions of what makes a river a river. Despite these obstacles, a struggle to restore the Rio Grande is taking root along its banks, aided by a diverse and committed assortment of NGOs and their allies in the various governments that claim a role in the river.

Pending litigation over the fate of an endangered species highlights the conflict between the public and private interests in the river, and in water in general. The Rio Grande is effectively controlled by the owners of the waters that it contains, but the struggle to preserve an endangered species has led to the reassertion of public interests in the river.

[Originally from jurist.law.pitt.edu - no longer online]

The United States and Mexico share the water of the river under a series of agreements administered by the joint US-Mexico Boundary and Water Commission. The most notable of these treaties were signed in 1906 and 1944.

https://en.wikipedia.org/wiki/Rio_Grande#River_modifications