After I reworked my map, I have a big plain (roughly 1165500 square miles). Can I still credibly divide the plain into different kingdoms and cultures or would a big empire with few cultures inevitably be the consequence? The research I did was not encouraging (USA and China).
Do you know what you get with a honking big plain? Honking big rivers. All that rain water must go somewhere and once the aquifer is full the only option is an ocean. There could be lakes in there somewhere, but eventually there must be rivers, big rivers, and large rivers regularly define national boundaries (especially if they're hard to cross or bridge).
Consider Europe, where a combination of rivers and mountain ranges resulted in a large number of unique cultures that, with the possible exception of the Roman empire, didn't begin to merge until the invention of railroads. (And it could be argued that they're not merged today... but that has more to do with nationalism than rivers.)
History is long and empires fall
The North European Plain, which "covers Flanders (northern Belgium), the Netherlands, Northern Germany, Denmark, and most of central-western Poland" (Wikipedia). And it is actually only a part of the much larger European Plain, which continues to the east and south, into Byelorussia, Russia, Ukraine and Romania, and to the west, into France.
The European Plain highlighted on the map of Europe, with national borders faintly visible. Map from Wikipedia, created by user Jeroen and available under the CC-BY-SA-3.0 license.
A large empire may form. It may then fall, and form again, and fall again. Consider the long and varied history of China:
The mighty Zhou
/ʈʂou/empire (which anyway held only what we would call today north-eastern China) fell to pieces at the beginning of the 8th century BCE, and from the 8th century BCE to the 5th century BCE, the Spring and Autumn period saw something similar to the European feudal fragmentation, complete with numerous small wars between small autonomous polities; then came
The Warring States, which ended in 221 BCE with the expansion of the Qin
/[tɕʰin]kingdom into a new mighty empire, which re-united what would be eastern China today; under various dynasties the empire survived until the end of the 2nd century CE, when it fell to pieces after the Yellow Turban rebellion, resulting in
The Three Kingdoms of Wei, Shu and Wu, which warred thoughout the 3rd century CE; in the end they were all conquered by the new Jin
/tɕin/dynasty. The newly re-formed empire had a very brief existence, because it soon fell to pieces resulting in
The 5th and 6th centuries saw China divided into the Northern and Southern dynasties, which were eventually reunited towards the end of the 6th century; the newly re-formed empire endured until the beginning of the 10th century, when it fell to pieces resulting in
Was conquered by the Mongols, piecewise, during the 13th century.
JBH and AlexP have already pointed out the European Plain, and ohwilleke has pointed out that the overall political situation won't be stable (which I agree with 90% - but I'll get into that). I'm just going to point out some things that will impact the overall physical geography and the geopolitical implications these impacts will have on your setting.
Do you have a young mountain system nearby?
The plains of the Argentina, the US and Canada neighbor the Andes and the American and Canadian Rockies, respectively. The European Plain is right next to the Alps; and the flattest areas of China, India, and Central and Southeast Asia are all next to the Himalayas. These are all relatively young - meaning post-pangea's-breakup - mountain chains.
Now, do you have to have a young mountain chain? No. Australia and the Sahara (which used to be a plain) both don't. But over time, a young mountain chain will affect the rivers in ways that could affect the politics of your region.
First of all, your plain is going to rum parallel to these mountains, with the highest stretch of your plain being along the foothills and the lowest part being on the exact opposite side. In the US, the highest part is along the Rockies, and in Europe its along the Alps. The lowest parts of these two plains are the Mississippi Basin and various seas, respectively.
This is because of erosion. These chains erode massive amounts of sand and sediment onto their neighboring plains over time. As the rivers carry the eroded sediment down onto the plains, they dump the sediment as soon as they slow down enough. This has the obvious effect of causing the altitude of the plain to slope upward toward the mountains and foothills.
Secondly, the erosion also affects whether the rivers are navigable and/or fordable. In extreme cases, your major rivers could turn into inland deltas, similar to swamps. You mentioned North America, so the example I'll use is the Platte River, one of two major rivers crossing the Great Plains from the Rocky Mountains to the Mississippi (the other being the Missouri).
The part of that wiki page that's important is the first paragraph under "Main Stem."
The first white settlers in Nebraska (which is named after one of the Native words for the Platte) described the river as being "a mile wide and an inch deep." That was not hyperbole. In its natural state, before the Army Core of Engineers dredged a deeper, main channel, most of the the Platte in Nebraska had both a heavy sediment load from the Rockies and an extremely low slope of 1000 vertical feet over 1,425,600 horizontal feet, starting at North Platte and going toward Plattsmouth. This resulted in a slow-flowing braided river, choked with sandbars, that ranged from a quarter mile to two miles at its widest point.
Unlike the rivers in the pictures that links to, plants actively colonized the sandbars in the Platte (Grand Island, Nebraska, is named after one such sandbar) and wildlife followed. So what the people living there had was a nearly impassable but resource-rich sand-bottom swamp.
You might note that because of snowmelt and the fact that the aquifer rises in the spring, the Platte still has significant flooding events around every fifteen years (although a long term drought combined with low-snowpacks in Colorado has kept it from happening for the last few years). At these times, the river can rise as much as twenty feet for a few weeks, and any plants without deep root systems get scoured.
The otherwise dry environment that surrounds the Platte makes it somewhat unique, but its morphology is not. Before it was drained, the European Plain had a massive swamp in the Dnieper River's drainage basin,
which is why Ukraine was Stalin's wheatbelt.
To sum this point up, if you have a young mountain system nearby, the plain will be higher along the base of the foothills and lower along the opposite side. Your rivers will bisect the plain by running from the mountains across to that opposite side, and if you have at least one major river there's a good chance it'll be a swamp.
The geopolitical implications of this are that the Platte formed a natural barrier between the north and south Plains, while keeping the people of those areas far enough away from each other for its water, arable land, and wildlife to be shared resources. Its edges also formed natural east-west trading routes as people went around it. Alternatively, do your societies have the engineering ability to drain or create bridges around the swamp?
Speaking of aquifers... Do you have an aquifer? If so, is it pressurized?
There are actually two types of groundwater - the surface groundwater that rises in the springtime, which can be an aquifer if it's deep enough, and the pressurized aquifers. You can see how pressurized aquifers work here:
The important thing to notice is that these aquifers are sandwiched under a layer of impervious rock. In the case of the Great Plains, that's a layer of shale that's underneath the sand that the upper aquifer is contained in. Just like the plains, it also slopes up toward the mountains from the Nebraska/Iowa state line, and it almost reaches the surface near Interstate 25 in Colorado. When snow melts in the mountains, in not only flows down in rivers, but some of it gets trapped in another permeable level under the shale. It continues flowing down, however, toward the east. But being trapped and forced down under the shale pressurizes it, and it wants to be at the same altitude as where it was first trapped in Colorado. So when you manage to drill through the shale, you get a gushing fountain that, in its heyday, was hundreds, if not thousands, of feet high.
People are naturally drawn to pressurized aquifers. It's just something about seeing a fountain come out of the ground, and of course it helps with plumbing too (that's why apartments and the sixth floor and below in New York City always have water - that's the altitude the reservoir is at, and the water wants to return to that altitude). If you have a pressurized aquifer and any of your political states are able to drill down to it, the fountains and the technology that created them will be desirable resources that they might fight over.
Speaking of shale and sand... What kind of soils do you have?
Aka, what kind of agriculture and ecosystems can your plains support? Obviously, this will be closely tied in with weather. It's kind of hard to tell on this map
but just look at how much of the European Plain is red, versus how much is green. (The light blue sandwiched just above Ukraine, that stretches off to the east, is the medium-yielding, highly resilient #3. The light blue that is mixed in with Poland, Germany, and Denmark is the higher-yielding #2.)
This is largely due to how far north the European Plain is. Large swathes of it would naturally be covered in boreal or deciduous forests that create poor soils. Again, the former swamp where the Dnieper flows through Ukraine is the only place where enough nutrients were deposited over time to create the most nutrient-rich soils. Until modern forming, this restricted what kind of agriculture could take place in different locations in Europe.
In the Great Plains, the natural soils are not much better. We have sand in the north, flint in Kansas, and clay everywhere. Much of the Plains are only suitable for grazing livestock... except for wherever the native fauna and flora enriched it. The natural ecosystem of the Great Plains is complex, with several engineering and keystone species. But over millennia, the mutual relationship between bison, prairie dogs, wildfires, and tall grass species enriched the soil with nutrients and protected it with a layer of sod. Once we had reliable access to water, this became the most productive farming land in the entire world. But this resulted in armed conflicts between the newly arriving farmers who wanted to privatize land and the established ranchers who wanted to keep it open.
The kind of agriculture your societies practices will influence their relationships with each other. Ranching societies will always be under pressure to have vast amounts of land available to their herds, plantation societies will always be under pressure to find new land to privatize for future generations' to establish plantations on (a system that was later adopted by ranchers), and farmers will want to keep livestock out of their crops. Everyone will want to maximize the amount of land available to them.
Last but not least...
Your plain will be a major trade route, and it will see a lot of warfare. There's almost no avoiding it, because its flatness makes it the easiest terrain to move over. Part of the Silk Road stretched over the Central Asian plains, north of the Caspian Sea. The Northern European Plain was the route armies took both east and west in Europe for centuries.
However, you can still develop distinct cultures on it. You might want to consider Russian culture: situated on the boundaries of both the European and Central Asian Plains, it is heavily informed by being at a crossroads of East and West. Or, you might want to study the Plains Indians
and Germany's historic principalities.
And from the sound of it, you've made your plain large enough to accommodate many types. Maybe they all have a shared past with a common conqueror, maybe they have a high degree of cultural transmission due to trade routes, but I'm sure they can all have unique ideas and products to call their own.
And finally finally...
There's nothing stopping someone in your setting from working their butt off to keep the peace. They don't have to be good person. They don't even have to be particularly nice or well-intentioned. It happens. I recommend this video, because Matt Colville explains better than I could.
Can I still credibly divide the plain into different kingdoms and cultures or would a big empire with few cultures inevitably be the consequence?
Both are a possibility, but societal organization always trends towards empires as one group gains or loses power and resources...and due to entropy any big empire will eventually fall because either the climate changes (rivers drain?), scientific advance happens that shifts values of resources or an incompetent leadership ends up leading the empire, thus leaving it crumbling as smaller regional powers decide to keep their own money and regulation instead of having it be a part of a failing system.
Here is a bit of a longer explanation as to why that would happen.
Terrain does not always have the same resources divided equally among the land. Some places have more resources, which will naturally attract more humans.
Rivers are good for as a water source for drinking and farmlands. They provide clay. They provide transportation and should they happen to connect other human cities together they can become a trading hub and earn large profits just by being a mediator group.
Forests would enable raw resources like wood, animal pelts, herbs etc.
You can imagine that people would settle on these hotspots of resources and develop their communities there. Since some resources would be more valuable then others those communities would get ahead in wealth and would be able to either achieve annexation or control of another community via trade and great relationships or the more straightforward way - by affording good weapons and armies.
More power a group has, faster it will grow because it will keep having more resources available to exert it's dominance.
However, resources are depleted. Climate changes and terrain does with it. What used to be an amazing spot full of resources 100 years ago may be worthless today. Scientific progress may make some resources obsolete or may new uses for existing raw resources which will just increase it's value tremendously, thus leading to rise of new powers or bolstering of existing ones.
Great leaders, either charismatic or brilliant strategists can change the political landscape as well.
Now, what could hold a big empire together? Number 1 reason is always economic benefit. As long as people are benefiting from being a part of the empire (or at least as long as the upper classes do that control others) they will remain part of the empire.
Another reason that may help is culture and religion. Similar people tend to stick together and help each others out.
Third reason is powerful military that makes the cost of secession from an empire to heavy to pay for most groups.
TLDR: Local powers would pop up, gain more power and will eventually form an empire. Empire will last for a few hundred years at best and then fall apart due to changes in economy, science, ideas etc