Mmmh, this is a question that I think is difficult to answer and is grounds for a very biased response, Im going to try to answer the question from the Mexica (Aztec) side of things and hopefully someone will answer the Spanish side, Im retelling from Sahagun and Tezozomoc (https://archive.org/details/cronicamexicana00alvaiala) who mostly speak about the conditions in Tenochtitlan and some demographic data I have on hand:
So lets begin with life in the Aztec "capital" of Mexico-Tenochtitlan:
This post by /u/400-Rabbit I think gives an excellent overview of life in Tenochtitlan: https://www.reddit.com/r/AskHistorians/comments/1uxdke/just_how_much_do_we_know_about_the_daily_life_of/cen94pd/
If I might add and point to some details:
We know the Aztecs had some kind of public education system Calmecac schools for pipiltli (kind of "noble" but not really until very late into their history) and Teposhcalli for macehualli (kind of "commoner" but not really), though neither was specifically barred from the other as Sahagun mentions some mazehualli attending Calmecac.
We know they had herbal and drug medicine as Sahagun notes that "The good medic (female) is knowledgeable of herbs, roots, trees, stones and has much experience in their use... not ignoring many secrets of medicine" and a very detailed knowledge of the human anatomy as Alfredo Lopez Austin notes in his book "Textos de medicina náhuatl" that aztecs had words and knowledge of most of the body's muscles, organs and bones.
We know they had a working judicial body in the form of judges and arbiters as Sahagun notes "The good prosecutor favors a band for whom they appeal having power (to sentence) and receiving a salary for it".
We know they had heavy trading as Tenochtitlan is noted by Cortez (Im taking this from the above post by 400-Rabbit) to have markets "twice as large as that of the city of Salamanca, surrounded by porticoes, where are daily assembled more than sixty thousand souls, engaged in buying and selling ; and where are found all kinds of merchandise that the world affords...."
Diet must have been varied Sahagun notes in his writtings merchants for: corn, beans, seeds, aji, tomatoes, cacao, pumpkins, honey, pulque (alcohol), chiantzotzol (kind of like white lentils), jicaras, fruits, fish, eggs, salt, birds flour, bread, and meats as is noted that there were sellers of dishes and atolli (its kind of a broth) and stews: "Pots are sold, made of (within them they have) pepper, tomatoes mixed with: aji, seeds, , tomatoes, green pepper, big tomatoes, amongst other things they make suculent dishes, they have as a job the selling of fried meats, chilmolli, and meats roasted beneath the earth".
And the city is said by Tezozomoc to have been cleaned regularly, as "people would come out in the mornings and sweep the streets in the mornings"... also there's this story about a particular tlatoani using the water works to flood and dry the city so as to "clean" it throughly after a prticularly nasty celebration... so I guess they could also do that and had working sewage, though nothing like modern individual house sewage.
Demographics
Now I dont want to say too much on this point as much of it is going to be wild speculation but here goes:
At the time of the conquest of Mexico in 1519, the population of New Spain (which at this point would be parts and pieces of the recently conquered empire) is calculated to have been at around 20 million inhabitants, with the capital itself of Mexico-Tenochtitlan having somewhere between 200,000 and 500,000 inhabitants (I personally take it as 300,000 to make some middle ground, for reference I often see it compared to the biggest European city at the time: Paris with some 300,000-ish inhabitants), this population we know is kind of standard for the region as the Mayan regions further south must have housed similar populations during its heyday (https://actualidad.rt.com/actualidad/262932-descubren-60000-estructuras-mayas-guatemala), so I believe those numbers are well within the probable.
Now as we have no real hard evidence on the above number I implore you to take it with a grain of salt, what we do know is that soon after the conquest a plague spread amongst the native population of the region killing between 8-9 million people, so at the very least we can say there were between 8-9 million people living in the region for sure soon after the conquest.
Now, and this is a personal observation, but from an 8-20 million margin in the approximate territories of the Aztec Empire in 1519, we to a population of 5-6 million a few years before the Mexican War of Independence in 1810, bear in mind this is for the entire territory of New Spain stretching through most of modern day Mexico into modern day US, which to my opinion, denotes that the living conditions in those territories, at least apparently, was reduced during the times of Spanish colonization.
Again this is a lot of speculation on my part, and I dont want to go very deep as this are approximates, not hard numbers Im posing, so while I do want to point it out, I would say you should take it with a grain of salt, especially considering Im Mexican and I have a personal bad opinion of the Spanish colonial period so I might be omiting information or agrandizing numbers in this regard.
Edit: Someone mentioned Michael E. Smith on the comments, it is now deleted, but Im now checking some of his articles and it seems legit, though Im not yet familiar with his work, and he does seem to be using the area of urban construction while ignoring the neighboring agricultural zones that existed around the cities, but his articles are at least using a sensible aproach to his results so Im tentatively posting them here for further reference: