This is a very interesting but very complicated question. There aren't really a lot of easy ways to approach it. The word citizen (as opposed to subject) implies that the person is a kind of stakeholder in the society. A citizen actively participates in the society and has an interest in its well being. A citizen identifies as part of the society. So the way I interpret your question, you're basically asking whether commoners were seen as stakeholders in Mesoamerican states. Whether a person's identity was constructed as a citizen in a city-state, or whether their relationship with the city was more of a sovereign-subject relationship.
Questions about commoner identity in Mesoamerican political systems are brushing up against the edge of what we actually know about these civilizations. Historical sources tend to focus on the elite aspects of society. Archaeology can help tell us a bit about what these people were doing, but we don't get the opportunity to ask them their opinions. Frankly, there just isn't a lot of information. Luckily, this is a topic I happen to know something about so I can at least give you a partial answer by telling you how commoners fit into the political system. You'll just have to draw your own conclusions after that.
Village and Neighborhood Level Politics
Mesoamerican governments had many nested levels, much like our governments. The smallest-scale level was a neighborhood- or village-level political unit that I'm going to call the barrio (using the Spanish word). The Aztecs called it a calpolli, the Maya called this a cah, the Tarascans called it a vapatzequa, and so on. There were significant cultural differences between each of these, but they're all variations on a central theme. Basically, the barrio was both a political and economic unit. It had its own leadership and rules for deciding membership. (The rules varied cross-culturally - in some cases membership in the barrio was determined by kinship, in others it was more corporate.) They organized community labor and often regulated land use. They paid tribute to a lord, but the lord did not own the barrio's land. There were often community shrines, and the duty to care for them fell to the barrio. In some cases (namely the kinship-based barrios) the barrio's leadership would have a role in marriage as well. When Mesoamerican armies were called to battle, the organization of military units was often by barrio. Every commoner that was not directly attached to a noble household was a member of one of these barrios, and they had an active participation in its affairs.
City-state
The basic political unit in most of Mesoamerica was what we often call a city-state (and again, different names for different cultures). This was usually just a city (or large town in some cases) and its surrounding farming communities. So by definition, even small city-states typically had more than one "barrio" as it likely included multiple farming communities. The city-state exacted tribute, handled diplomacy and war, organized large scale religious celebrations, and held a market on a regular schedule. It was through these activities that commoners interacted with the city-state government.
Their participation in state government, however, was much lower. Mesoamerican societies had powerful aristocracies. There were some city-states where the government was more of an oligarchy than a monarchy, but even in these cases the nobility dominated the political realm. There are very few examples of a "middle class" in Mesoamerican societies.
Now that's not to say that commoners weren't stakeholders in the city-state. They participated in markets and in public festivals that were held at the city level. Many city-states had a patron god and a kind of city-state identity could be formed around that. Additionally there were several examples of institutions (military orders, for example) that a commoner could join which gave them more stake in the political system.
The Empire
Often in Mesoamerican history their were political bodies that emerged that superseded the city-state. This happened when one or more city-states managed to gain power over its neighbors. With one notable exception (the Tarascans - the coolest of the Mesoamerican empires) these "empires" did not usually interfere on the political affairs of the city-state. Instead, they usually left local political systems in tact as a sort of vassal or client state. When they did so (again with the exception of the Tarascans), they exacted tribute from the city-state or community as a whole, and did not tax the barrios or the commoners directly. In some larger empires, they may set up a provincial "capital" that handles tribute collection for a larger region, but for the most part the idea was to avoid interfering with things in conquered cities unless it was necessary.
This meant that the politics of empire building, in a day-to-day sense, was not something that commoners interacted with frequently. For the commoners in a capital of one of these empires, they may have an active stake in seeing their city continue to dominate its neighbors. Wars of conquest can provide opportunities for social advancement, and the influx of tribute will mean bigger festivals and more trade. But for commoners in a conquered city-state, being part of an empire probably wasn't a good thing. It meant the nobility of the city had to pay tribute to another city, and that meant the commoners had to pay more tribute to the nobility.
Identity
So what does all this mean about how your average person thought of their "citizenship?" Basically the society that a person belonged to first and foremost was the local barrio. This is where the bulk of their political life existed. If they had a polity that they belonged to beyond that, it was the city-state. Their relationship with the city-state government, however, was more of a sovereign-subject relationship than one of more active political participation. Larger political structures like empires, when they existed, were external political forces that the commoners interacted with marginally.
Now, can we say that this is a definitive explanation for how Mesoamerican cultures defined citizenship? No. There's a lot of subtly and nuance to this that we can't get into here because it would be longer than a reddit post, but also because there's not a lot in the way of "insider's" perspective on this. Identity is a subjective phenomenon that's hard to get at with an outsider's lens. I bet if you asked 20 people in any modern country what they think citizenship means, they'll probably give you 20 different answers. The same was true in the past.
EDIT: For those with JSTOR access (sorry), David Carballo gives one of the best syntheses on the research of commoner households that I've ever seen. In it there's a section on neighborhoods and their role in Mesoamerican culture.