In Italian, azteco seems to be a perfectly regular adjective.
What precedent is there for preferring one or the other of these in Latin? The etymology says that Spanish azteca comes from a Nahuatl suffix -tecatl. Perhaps people then treated the Nahuatl suffix grammatically by analogy with the Latin/Greek suffix -thēca/θήκη. Coming from classical roots, this suffix only forms nouns, like bibliotheca and discotheca, not adjectives, so the analogy is imperfect, but that's often how it is when extending precedents to new cases.
Another reasonable possibility is to reject the analogy as false to begin with, since the Nahuatl suffix means "inhabitant", whereas the Latin suffix means something more like "container for a collection". But it's also reasonable to think of an ethnic group or civilization as a sort of contained collection.