Well, it's hard to make direct comparisons with European nation-states or the globalised alliance networks of the modern world.
But I think one might at least argue that there was a sort of rough analogue to NATO and Warsaw. You're probably already familiar with one of the actors - the Aztec Empire.
In popular depictions the Aztecs are often depicted as monolithic, but in Nahuatl they called themselves the 'Triple Alliance' - a union of the three city-states of Tenochtitlan, Tlacopan, and Texcoco.
In the 13th century the various Nahua city-states in the area around Lake Texcoco existed in a constant flux of shifting alliances and small-scale wars - think Classical Greece. Over the course of 200 years, one city, Azcapotzalco, came to dominate its neighbours, establishing the Tepanec Empire.
A Tepanec imperial succession crisis saw the new emperor crack down on the cities that had supported his brother; the king of Tenochtitlan was captured, humiliated, and possibly murdered on his orders, while the king of Texcoco was forced into exile. These cities went into open rebellion, were joined by the city of Tlacopan, and emerged victorious three years later.
The victorious allies formally cemented their relationship, forming the nucleus of what we now know as the Aztec Empire. I draw the (very loose) comparison with NATO because each city-state retained its own leadership and army, but the three went to war as a unified front and divided up the spoils amongst themselves.
While the Tepanec Empire had been limited to the Nahua city-states of the Valley of Mexico, the Triple Alliance expanded aggressively into foreign lands, attacking the Mixtecs and Zapotecs to the south, the Totonacs and Huastecs to the east, and (of particular interest) the Purepecha to the northwest. But the empire was highly decentralised, with these conquests generally being left more or less autonomous as long as they paid their tribute and provided their armies when demanded.
The Triple Alliance was never an equal partnership. The junior partner Tlacopan received one-half the share of tribute of its two stronger counterparts, and Tenochtitlan was always the dominant decision-maker, with this sort of 'hegemony of hegemonies' being formalised after 56 years. To lean on your comparison, it's a bit like how the USA is one of many independent member states of NATO, but NATO without the USA would be leaderless and functionally gutted.
To keep this whole shaky comparison tottering along, the Soviet Union to the Aztec NATO would probably be the previously-mentioned Purepecha (or Tarascans), the most interesting pre-Columbian empire that nobody's heard of.
Living in northwestern Mexico, they were among the most successful rivals of the Aztecs, beating back an invasion in the 1470s and fortifying the borderlands.
Originally a typical tribute-based hegemony based out of the city-state of Tzintzuntzan, the later Purepecha Empire became highly centralised (for its day), with the state ultimately directing matters such as infrastructural projects, the leadership of craft guilds, the displacement and/or settlement of subject peoples, and marriages among the nobility. Like in Europe, the Tarascan king owned his own private personal lands, and unauthorised poaching or use was a grave crime. The Tarascans also routinely replaced the nobility of conquered cities with their own governors.
Unfortunately little has survived in the way of evidence or records compared to the Aztecs, but what remains paints a fascinating picture. User Ucumu has written some brilliant stuff on the Purepecha Empire that's well worth a read.
When it comes to other rivals, the fellow Nahua city-state of Tlaxcala was a famous enemy of the Aztec Empire - the two engaged in near-continual warfare for nearly a century, with the Aztecs never capable of breaking Tlaxcalan resistance, but better able to weather attrition over time.
When foreigners unexpectedly arrived from beyond the horizon, the Tlaxcalans (after some initial tension) recognised their potential as game-changers and allied with Cortes. With Spanish assistance, the majority-Tlaxcalan army was successful in breaking the back of their despised Aztec rivals and devastating Tenochtitlan, ultimately helping to pave the way for Spanish domination of Mexico and the near obliteration of native Mesoamerican civilisation.