Kind of. "Aztec" is not an ethnicity; it was a political confederation that ruled parts of modern Mexico for about a century leading up to the arrival of the Spanish. Let's establish from the beginning here that the people who lived in the Aztec Empire would not have self-identified as "Aztecs," which is a term invented centuries later (the actual endonym was "Mexica"). They would have identified themselves by their city-state.

A significant percentage of the native Mesoamerican population died of disease and war after the Spanish arrived. The current population of Mexico is made up of descendants of the native population and European colonists. There has been a lot of mixing over the past five hundred years; few Mexicans today are pure blooded native (or European). So in that sense, yes, the modern descendants of the people who lived under the Aztec Empire primarily live in Mexico.

But it's also important to know that modern Mexico is a much larger country than the Aztec Empire was, and it includes a great deal of land that was never ruled by the Mexica. There were native people living in those places five hundred years ago who had nothing to do with the Aztecs. The Yucatan Peninsula, for example, was the center of the Mayan civilization. The modern descendants of the Maya live in Mexico as well, though they are not Aztec.

TL;DR: Modern descendents of the Aztecs are (largely) Mexican, but there are a lot of Mexicans who are not descendents of the Aztecs.