To tackle first part of your question (medium of exchange) first, while there was not an official currency, there was absolutely standards of exchange. The commonly used "currencies" were cacao beans and cotten cloaks (quachtli) with the exchange between the two items commonly cited as 65 beans to a cloak (though this would depend on the quality of the cloak.

Cacao beans were a common and popular small currency (which even had issues with counterfeiting), with cloaks acting as a larger denomination, but they were not the only common items of exchange. The barter system that dominated the markets of Mesoamerica meant a trade was a trade, but copper "axe money" and gold dust were also common currencies. We can see this quite clearly in the tribute for the Quiauhteopan area (in the SW of the present day state of Mexico). Twice annually, that area was to provide 400 quachtli and 80 copper axes, among other items. This was basically a demand for a pure cash payment, since these (unlike demands for gold dust, for example) were finished products and would not be further processed into other goods.

This brings us to the second part of your question, how to measure wealth. The primary thing to keep in mind is that Aztec society, at least by the late imperial period, had a fairly rigid social class system. While there was a sort of life peerage that could be obtained early on (cuauhpilli), through military prowess, this avenue to advancement starts to close with the reign of Motecuhzoma Ilhuicamina and is gone by the time the Spanish arrived in the reign of Motecuhzoma Xocoyotl.

The former Monty is also famous as setting out the official foundation of Aztec law, which included several sumptuary measures. Going by Duran's recounting of these proclamations, there were restrictions on who could wear certain styles of cloaks; who could wear cotton clothes; what material jewelry and piercings could be made from; and even who could build a second story on their homes. Basically what we see is a reinforcement of the traditional class distinctions in Aztec society.

What were those class distinctions? The simplest way is to see divisions between teuctli (upper/ruling nobility), pipiltin (nobility), and macehualtin (commoners), with mayeque (tenant famers/serfs) and tlacohtin (slaves) below those groups. Like all general frameworks though, the reality is that there was ambivalence and nuance among and in-between these groups. Hicks (1999), for instance, identifies a sort of middle class made up of 6 roles that fell at the intersection of the petty nobility and the wealthy commoner:

1. lesser political officials

2. stewards in charge of labor and goods

3. artisans in the luxury trades

4. merchants who brought exotic goods to the market and imported raw materials for the artisans

5. ritual and ideological specialists

6. enforcers and military professionals

Sumptuary laws being what they were, actual display of wealth in public was prohibited (though there is some archaeological evidence of displays of wealth inside homes). Instead, what we really see is a system of the bestowment of social favors and gifts which could translate into material wealth. The last category, that of "enforcers and military professionals" is the key example of this. Even though advancement up the social class through military prowess became less tenable, achievements on the battlefield could absolutely garner rich gifts in the form of ritual clothing (certain headresses); better material adornments (gold armbands) and the right to wear them; or even material wealth (several cotton cloaks).

The ritual and ideological specialists category is deceiving, in that it encompasses everything from service providers like diviners, teachers, or midwives to the actual priesthood. In the former case, those was more of an elite trade, and some of these professionals might find patronage in the employ of the nobility. The priesthood, however, functioned more like state-employees, supported by tax, tithe, and tribute. Being a priest in Mesoamerica was a full-time and highly demanding job, and the priests would live at the temples they served. This freed them up to perform auto- and exo-sacrifices; fast for months at time; and perform various other ritual behaviors.

There's a particular quote from Bernal Diaz del Castillo about being greet by priest entering Tlaxcala that seems appropriate here:

Next came the whole body of papas, of whom there were great numbers in the temple service. They carried the pans with glowing embers, and perfumed us. Some of them had on long white cloaks, after the fashion of surplices with capes, as worn by our canons. The hair of their heads was long and matted together, so that it would have been an impossibility to have put it in any shape or order without cutting it off: besides this, it was completely besmeared with blood, which trickled down over their ears, for they had been sacrificing that very day. The nails of their fingers were uncommonly long, and they held down their heads on approaching us, in token of humility. It was told us that these men were greatly revered for their religion. The principal personages now gathered themselves around Cortes' person, and formed a guard of honour.

The priest was... otherworldly. Revered and respected, yes, but their domain was not of material things.

In a way though, the idea of being revered and respected was a better socio-economic currency than actual currency. Since expression of wealth was restricted by class, they only way to display fabulous goods was to advance up the social ladder or to receive special privileges of display from the ruling class. To return to Hicks, he identifies a few ways in which the "middle class," particularly those first 4 categories of bureaucratic nobility and elite merchants/artisans achieved status and wealth:

• Proximity to the Seats of Power

• Gifts of Valuables

• Market Income

• Visible Signs of Merit

• Exemption from the More Onerous Obligations

• Extra Land and Help to Work It

What we are looking at then, is a system wherein social access to the upper elite could provide a venue for have partial access to the privileges of that class. The tribute system acted as a giant material transfer of wealth from conquered areas into the core Aztec cities, but the distribution of that wealth was highly dependent on the social obligation and personal whims of the state. The state, in this case, being the dynastic ruling families of the altepetli. Actual material gain through economic activities could allow for access to this wealth though personal relationships with the ruling class, but, in a sort of positive feedback system, could also allow for greater opportunities for economic advancement and the lessening of things like tax or tribute burdens.

The land aspect is particularly interesting. As a general rule, only the nobility personally owned land in the Aztec system. The rest was held in trust by the state at the calpulli (neighborhood) level. Each family was granted a plot of land and, so long as they productively worked it, it was effectively theirs. Abandonment though negligence or absenteeism meant the land could be reclaimed and given to another family. Kellog's Law and the Transformation of Aztec Culture, 1500-1700 covers this dynamic very well, particularly as the colonial era starts to generate a plethora of legal documents in Spanish and Nahuatl.

To lope towards a conclusion, wealth and status could be accrued in Aztec society through the participation in the state bureaucracy, luxury good trade/crafts, military success, and/or participation in religious ritual/tutelage. Actual wealth had a variety of standard media of exchange, but this did not preclude other common wealth goods like precious stones, exotic feathers, basic goods, luxury foods, etc. While wealth could provide personal access to elites, it was only through those elites that wealth could be openly displayed, and also only through them that personal ownership of land could be granted. So while physical items were certainly an important factor in accruing wealth, there also needs to be an acknowledgement of the social mores and actual laws which restricted access to both goods and behaviors.

It's hard to say because we don't have a lot of written records from mesoamerican cultures and the ones that we do were written after the conquest in supervision. From material I've read about Aztecs, the wealth would be tied to your social status. In a sense that Aztec society would place great importance on their men being skilled in art of war who have proven themselves in the battle multiple times. Most Aztec society were peasants, some merchants (who acted as spies a lot of the time), warriors and upper classes that were mostly hereditary (comprising of less than 10% of their population). Since they've placed great importance on bravery in the battle, it was absolutely possible to raise in the ranks and become high status "citizen" after several campaigns. They could get land, marry multiple women, to name a few perks. It became less common during later times of empire and elite status became hereditary. But that's not to say that warrior status became less desirable, it became even more so. For example, if you had your main heir to inherit your wealth and he was cowardly in the battle or couldn't capture an enemy after several campaigns, he would have his head shaven and thrown out of the warrior class, virtually making him a peasant who would have to farm or do other low skilled work. And if the bastard child on the other hand proved himself on the field then he would gain all the favor. The pressure was huge actually and individual bravery was emphasized. Aztecs actually wanted to capture as many people as they could rather than kill them (they did that later, ritually).

Just an interesting fact about Aztecs. Food and feasting was a big thing for them. Which brings me to the next point. Merchants did use cacao beans as currency. Chocolate was a big thing, they made a drink out of it that was offered to gods. And another way for elite to stand out was to make huge feasts. If you could do this you were seen as grand and if you couldn't then you weren't. Allegedly, Moctezuma had 300 dishes and 1000 prepared for the royal household, daily. They had an interesting relation with food. Women were supposed to feed men, and men were supposed to feed the gods with the blood of the captured. The whole ordeal with maize and preparation of it was fairly ritualized because it was so much important.

So I guess this is a bit of a not so straight forward reply to your question but I hope I've showed or gave you an idea how Aztecs would display wealth and what they valued.

Sources: 'Cost of Courage in Aztec Society' by Inga Clendinnen; 'Que vivan los tamales' by J. M. Pilcher