I can't speak for the Aztecs, but the Athenian Empire is definitely called that by modern scholars. Two things complicate the matter: (a) the Delian League is not quite the same as the Athenian Empire, and (b) the Greeks didn't have a word for 'empire'.

To start with (a), the Athenian Empire evolved from something we call the Delian League, which was not an Athenian Empire. It was an alliance of equals established between Greek states in 478 BC, after the Spartans relinquished leadership in the ongoing war against Persia. This alliance had its treasury on the sacred neutral island of Delos, which is where it gets the name "Delian League". From the outset, Athens contributed the largest number of ships to the alliance, and they may have effectively controlled its strategy, but it didn't in any sense own or rule the other members.

However, over time the Athenians gradually increased their control over the rest of their alliance. When states tried to break away from their obligations to the League, Athens would attack them instead of letting them go free; the result was usually that the breakaway state had to give up all its ships, tear down its walls, and resume paying tribute into the League treasury. Meanwhile, Athens encouraged other states to contribute money rather than ships to the League. The allies were happy to do this, since it meant they no longer had to send their own men out against the Persians - but the result was that they became dependent on Athenian naval power, which was funded out of the tribute they paid.

For the first 25 years or so, it's arguable that the Delian League remained a league of willing allies led by Athens. But by the 450s BC it seems this was no longer the case. Athens started to impose democracy onto rebellious allies, thinking that such regimes would look to Athens and be more loyal; they may have forced the entire alliance to adopt Athenian coinage and measures. Of all the allies, only 3 states besides Athens (Lesbos, Chios and Samos) still contributed ships of their own. In 453 BC they officially moved the League treasury from Delos to Athens. This was officially to safeguard it, but it should be properly understood as a power grab; from this point on, no one would dispute that the alliance was an unequal one, with all the lesser states subject to Athens.

In short, the Delian League wasn't called the Athenian Empire because the distinction is important; it marks a shift in power balance that was only completed some decades after the League was founded. However, some scholars see the League as an Empire from the start, while others continue to refer to the Delian League right down to the Peloponnesian War. The point of transition is not precisely known, and people disagree on when it was.

This is made more difficult by point (b), the fact that the Greeks didn't have a word for empire. They weren't able to express clearly that the Delian League had transitioned into an Athenian Empire since no such concept really existed. Greek texts only refer loosely to the Athenian arkhê (dominion, sphere of influence) or simply to "the Athenians and their allies". There is some evidence that the Athenians stopped thinking of the other states as allies and started calling them subjects, but they did so inconsistently and the usage doesn't mark a clear shift in how they saw the alliance. Instead, right down to the end of the Peloponnesian War, the Athenian side of the conflict is that of "the Athenians and their allies" - not subjects - and no new words are invented to express what this thing is that the Athenians now possess.