Early European settlers along the Animas River in northern New Mexico thought they'd discovered Aztec ruins. So they named their town Aztec. However, the Ancestral Puebloans (who we used to call Anasazi) who built the stone villages were only very distantly related to the Aztecs. They also predated the Aztecs. But mistakes often stick.
Archaeologist Earl H. Morris rebuilt the kiva in 1934. There was a lot of guesswork involved, so it's more about an impression than architectural detail. And about getting out of an afternoon cloudburst.
A fellow visitor asked if I knew what this was. Her guess was as good as mine. And Earl H. Morris's.