I feel like somebody with even a passing interest in the Aztecs quickly learns that the term that's commonly used to describe is something of a misnomer that wasn't anything like what the people themselves described themselves as (similar to the 'Byzantine Empire').
My understanding is that 'Aztec' is derived from the mythical ancestral homeland of the Mexica but they didn't describe themselves as such, and they were even directly instructed to not refer to themselves as Aztecs. In addition the conception of the empire as Aztec tends to elide the fact that it was a confederation with two other peoples.
With this kind of thing in mind, how exactly did it become normalized to refer to the Aztec empire in the way we do now in English? Why don't we call it the Triple Alliance or even something like the Mexica empire? The latter has its issues but it still seems a bit more accurate compared to the Aztec Empire.
So, let me break some things down which will hopefully make things clearer.
The term Aztec is a more recent term created by Alexander von Humbolt to refer to multiple Central Mexican ethnicities whose history includes a mythic-historic journey from an ancestral homeland to the Basin of Mexico. Aztec does not refer to a single ethnic group, but rather a collection of ethnic groups all residing in the same geographic area at the time of Spanish contact. One of those groups was the Mexica who eventually founded Tenochtitlan. And as you pointed out, they were in an alliance with two other city-states: Texcoco, which was ethnically Acholhua, and Tlacopan, which was ethnically Tepanec. These three city-states together conquered other city-states both within and outside the Basin of Mexico.
We refer to this alliance as an empire not because of the three city-states working together for mutual defense, conquest, and commerce, but because these three cities sat on top of a hierarchy of other subjugated polities in Mesoamerica. They formed a tributary polity in which other cities and states, once conquered, owed yearly tribute to the cities in the form of food, crafts, or labor. Cities in the Basin of Mexico tended to provide food because it was easier to transport bulky foodstuffs shorter distances, while cities outside of the Basin of Mexico tended to provide crafts (which were high value, low number) or labor (often auxiliary soldiers to help conquer adjacent areas or to pacify unruly territories).
One reason to continue to refer to this alliance as Aztec rather than Mexica, in my opinion, is because the term Aztec is artificial it does not exemplify one ethnicity from the alliance over another. To call it the Mexica empire would erase the contributions of the Acolhua and Tepanec ethnic groups who were vitally important to maintaining the tributary empire they had created. Aztec also allows archaeologists and historians to refer to a broader and wider geographic region in which multiple ethnicities shared many cultural traits and whose ethnic distinctions may be difficult to discern through material culture in the archaeological record. Unless particular ethnic groups had their own ethnic crafts people producing their own unique goods, everyone is purchasing and trading for the same ceramics, obsidian knives, and greenstone and shell jewelry in the marketplace as everyone else. Perhaps how they used those items differs between ethnicities, but to determine that would require a lot of data across multiple sites in order to discern those differences.
So is the term accurate? Not entirely. But is it still useful and does it serve a purpose right now until we are able to obtain the data necessary to make those finer grained distinctions? Absolutely. So I hope this was a helpful explanation for you.