For the first century or so Aztec government kept being nominally in control of several regions in the Valley of Mexico, though as individual vassals of the overall Kingdom of New Spain (and Spain by extension), rather than as a unified country, something that has to be understood, is that Mexico is a huge region, more so because most of it is rough terrain with most of it being criscrossed by mountain chains, so often the territory is more vertical than you would think from seeing it on a map, with this circumstances aztec "auxiliaries" and middle-men became invaluable for the early colony, many wars, for example the Chichimecan War (which Spain lost btw), or the Pueblo Rebellion were mainly fought with the use of conquered peoples pressed into service by the conquering spanish.

As an example of this, in General History of the Things of New Spain, Bernardo de Sahagun makes note of the Mexico City governors that followed the conquest, check if their names ring a bell:

(after the conquest) 20- The 12th governor (at this point, it changes from "lord" to "governor") of Tenochtitlan was named Andres Motelchiuh, and governed for 3 years in the time of the spaniards which saw the conquest of Cuextlan of Honduras and Anahuac (note: this is likely a mistake by Sahagun, as Anahuac is the nahwa name of the continent, not a country), after which he went with Nuño de Guzman (note: "went with" in this case would more likely be interpreted as "was forced to go with" by modern standards) to conquer Colhuacan were he died.

21- The 13th governor of Tenochtitlan was named Pablo Xochiquen and governed for 3 years.

22- The 14th governor of Tenochtitlan was named Diego Uánitl and governed for 4 years.

It goes on like that until around the 16th governor and then goes on to name other lord and governors from different cities.

This changes definitely around 1585-ish, when Spain has enough of a foothold to start governing the region directly, it is here when titles and lands are taken away definitely from indigenous nations and their lands become the direct property of Spain, the nawha, mixtecs and others mantain some level of autonomy for a time, though they will loose them over the centuries, weaker nations like the Tlaxcaltecas have their land outright taken from them in legal movements and trials that will last for a good chunck of the colonial period though that is another story.