The groups in the central country have a common cultural ideology that unites them and distinguishes them from the other three
Multicultural nations generally work by having a common cultural ideology anyone can be a part of regardless of social class or ethnic group. Having a common group identity that anyone can be a part of encourages social cohesion and prevents “us versus them” tribalism reminiscent of The Sneetches. That’s how the Achaemenid Persian Empire did it, that’s how the Romans did it (Romanization), that’s how ancient China did it (and to some degree still does today), that’s how the Soviet Union did it, that's how India kind of does it ever since they fought for independence from the British Raj, and that’s how many Western nations do it today (most notably but not restricted to the U.S.). Exactly how well these countries live up to the ideals set is something that literal volumes have been written over, but in bare essentials that's the idea of how it's supposed to work. It doesn't have to replace the parent culture, but has to at least complement it (as an example, look at immigrant communities in many nations across the world today).
What this would look like in your case is while members of these races across the four countries might look like they belong to the same culture, they wouldn't consider themselves as such. Lets say these are elves, orcs, and humans for sake of example, since that's the general analogy previous answers have gone with. Elves in the central country would consider themselves to be a separate people from elves in the elven country, and there would likely be cultural differences between them. The elves of the central country would consider themselves more culturally similar to orcs and humans from the central country than to the other elves (and likely vice versa), and their subculture within the broader central country culture would likely be a mixture between elven traditions from their own background, cultural ideas they picked up from diffusion with the orcs and humans, stuff they came up with on their own after immigrating from elf land, and the general central culture background ideals that hold the country together. If these groups cannot interbreed or rarely do so as your question seems to suggest, cultural barriers would remain up longer compared to all-human countries where intermarriage is possible.
Switzerland, as mentioned in another answer, is a good example of this. Despite having people of Italian, French, and German descent, people in Switzerland don't consider themselves Italian, French, or German. They consider themselves Swiss.
The people in the central country, at least as a majority, have to want to be together based on cultural unity to some degree for the country to work. Iraq, Rwanda, and many other countries in Africa and the Middle East shows what happens when this doesn't occur. In these cases various groups that didn't have a common cultural identity were divided up together into a single country by European imperialism (or just arbitrarily assigned one in the case of Rwanda) and the various groups butted heads. This has even been mentioned as one downside to the overthrow of Saddam Hussein when talking about the long-term stability of Iraq, the Iraqi military was creating a common Iraqi identity that was keeping the various factions together and allowing the country to function to some degree.
A good example of this in fiction is the United Republic of Nations in the Avatar series. Despite being descended from Fire Nation colonies set up in Earth Kingdom territory, the United Republic of Nations are clearly a distinct cultural entity from either country and this only becomes more emphasized as time passes in-series.
When the Fire Nation tried to give the colonies back to the Earth Kingdom after the Hundred Year War, the colonies rioted because a significant portion of the population had mixed ancestry and they were threatened with deportation, and the population did not see themselves as culturally identical as the Earth Kingdom. This is even more emphasized in The Legend of Korra, where the United Nation of Republics is a technologically advanced metropolis with its own methods of government different from the monarchial Fire Nation and Earth Kingdom, its own unique styles of the various bending arts, and even differences in fashion, whereas the Earth Kingdom is largely technologically backwards and impoverished and culturally very different (though mostly unchanged from The Last Airbender-era due to government mismanagement with the exception of Zaofu). What happened to the Fire Nation is never shown.
Religion can be another way to achieve this (I mean, religion is basically a form of ideology anyway). The Abbasid Caliphate ruled over a lot of different cultural groups, including Arabs, Egyptians, Persians, Seljuk Turks, and various Christian and Jewish groups, among others. One major factor holding the caliphate together was the fact that most of them practiced a common religion, Islam, and the government put in policies to try to convert people who weren’t Muslim through incentive of tax writeoffs and other means.