Most historical civilizations did study the civilizations that came before them. However, this wasn't necessarily "study" the way we think of it today. Civilizations often referenced older ruins and stories about previous cultures in their mythology and origin stories. Since most civilizations were not the first urban societies in their areas, they had ample traces and legends from the past to study.

The Aztecs are actually an excellent example of this. The Aztec Triple Alliance came to power late in Mesoamerican history, after dozens of other urban civilizations which had stretched back for thousands of years. Cities of Olmec, Zapotec, Toltec, Mayan, and many other Mesoamerican societies were built long before the Aztecs built their state. And the Aztecs were very much aware of some of these ruins. Perhaps most importantly, they considered the long-abandoned city of Teotihuacan an important site. Teotihuacan was one of the largest cities in the world during the early first millenium AD, with some 100-150,000 people, monumental architecture, and extensive influence over surrounding areas. The Aztecs considered Teotihuacan's astounding ruins to be the birthplace of the gods and the location where the current version of the universe began. So they were certainly aware of, and discussed, ruins of civilizations before them, even if the way they discussed history was different from how we do currently.

Almost every other civilization had some sort of interaction and study of ones that preceded it. The Inca studied and created ruins to further their own ideology, the Romans tied their identity to Greek history through the Aeneid, and countless other civilizations built their own legitimacy through studying and interpreting former societies. Perhaps the only civilizations which did not study previous ones were those which had no former civilizations to look to -- the first urban societies could not study civilizations which existed before them, since there were none.