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In my world there are multiple continents with many cultures. In a far away continent, there are kingdoms that exist there. I was wondering how could they be medieval style like the mainland, and if it is even possible at all.
Things to consider:
The question as asked is vague, and may want to be narrowed down, but here is my understanding of the question, and my best swing at answering; you appear to be asking 'What conditions would lead a medieval culture aesthetically similar to Europe's to develop independently on two entirely separate continents, and just how likely is that to happen?'
My answer - there needs to be large reserves of metals, especially iron but also copper, tin and gold, and a history of metallurgy. There needs to be a large and robust trade network, and domestic horses or similar animals. There needs to be a substantial winter in more than half of the available landmass. There needs to be a history of militaristic empire that pushes society towards designing standardized weaponry and protective structures like castles, walls and towers. As long as both continents meet these conditions without overlapping with one another, it feels realistic that both continents might have arrived at similar medieval eras at around the same time in a sort of parallel evolution.
However, it is important to understand just how diverse this image is, and has to be in order to exist in the first place. When you say 'medieval Europe,' you are including a vast world of differences. For nearly 700 years, muslim kingdoms controlled large portions of Spain; Italy was a mess of oligarchic and democratic city states fighting one another in wars mostly funded by trade with India and China and largely fought by mercenaries, some from as far away as england; Other areas might have been ruled by kings, or by popes, or by councils of nobles who elected their ruler; The boundaries by which we may now define 'Europe', 'Asia' and 'the Middle East' would have been drawn very differently, and those differences would be felt in the way people dressed, the design of their architecture, and the weapons and armor they used in combat. And that's just Europe - There are plenty of other parts of the world that had medieval eras, from Africa to continental Asia to Japan. Both of your continental areas must feature this level of diversity to be fully believable. There is no such thing as a medieval Europe without the backing of a larger world to compete with, trade with, to conquer and be conquered by.
Also, even if they are at a similar level of technology, their cultures will necessarily be different in big ways. Religions, languages, and value systems all will have developed independently. Certain things they will have in common - technology such as bows and crossbows, levers and pulleys, clockworks, as well as the use of gold as currency if it is available, all of these things have been known to develop independently in separate cultures, as they draw on natural facts (tension, gravity, and the fact that gold doesn't tarnish and thus retains its value). On the other hand, other technologies such as gunpowder, glassblowing or silk, depend on the presence of specific resources and the understanding of how to utilize them - and if one continent has one of these and the other doesn't, it will spell out a rather large difference. Even something that seems as small as what natural fibers and colors of dye are available to them will affect the way they look and feel.
In short, as long as you do some research and make your two continents feel enough like they have truly different histories, both in terms of the people and the animals and plants that live there, it can still feel realistic that both continents might feature medieval feeling societies.
Rediscovered.
https://www.deviantart.com/lpsdc/art/The-Lost-City-868020985
Millenia before the events of your story, there was an Atlantis-like ancient empire whose domain spanned the continents. There was a cataclysm and then a dark age. Memory of the empire faded into legend - mentions of their doings in holy books, cyclopean ruins testifying to lost engineering prowess, tales of far lands and exotic beasts.
The people of the other continent are descended from colonies of that empire, just as your main medievals are. They are "rediscovered" but their language has familiar aspects to your people and with effort they can be understood. Your two peoples share some gods and other cultural aspects. They are different but not wildly different. It has only been 1000 years.
Depends on how you define "like" ...
Speaking the same language, having exactly the same kind of technology, without knowing each other? Not really.
But consider how little China and Europe knew of each other, during Roman times. Both ruled their edge of the world quite effectively, both traded through intermediary ports (silk was a big import), yet direct contact was limited. They were on the same landmass, yet trade was by ship and no one ship would usually do the whole voyage.
Without the trading ports in India, they might have known as much of each other as the Romans and the Mesoamerican city states.
Q: "I was wondering how could they be medieval style like the mainland, and if it is even possible at all."
.. and recent common roots would require an implausible scenario: the art of navigation and sailing, with underlying knowledge of astronomy and ship building would have gone lost and rediscovered a few centuries later. Like 10 generations of amnesia, the 11th generation suddenly finding memories back ? Without such recent connections, any similarity to "home" seems implausible, when we take real Earth history as a reference.
In a few hundred years, civilizations overseas will diverge.
We're not talking antiquity, this is medieval times.
As pointed out in earlier answer, you could make up an Atlantis- or Prester John like scenario, that is the freedom of writing your story. However, the Atlantis-version would rule out similarities in medieval times, occurring thousands of years after Atlantis. In thousands of years, traditions will drastically change, religions will be introduced, technological advancements based on local resources will have occured. But these conditions and resources differ, everywhere.. Medieval European folks, when rediscovering Atlantis in 1100AD would find a completely different world, despite its origins were European.
No real world examples
Will a foreign civilization look like your world when you discover it ? In Earth's past it never happened. For instance the Dutch, while "discovering" India and Japan experienced how difficult it was to make friends and set up trade. The more complex the overseas civilization was found to be, the more brand new elements they would find. When the Spanish conquered parts of South America they did not even recognize civilization present.
Thule, lost Norse colonies, etc. The Histories and mythologies of Europe are filled with references to supposed kingdoms across the sea, or just out of reach, or just over the poles. Even modern fantasy has such things, like the elves departing to their lands across the sea.
Any group of people sufficiently motivated to leave your continent will carry with them the traditions of that land, and if they crossed the Atlantic (for example) and colonized, then they would establish a beachhead of your culture wherever they went. They could have brought traditional seeds and plants, preserving traditional foods.
But don't make the mistake of thinking they will be clones of your existing society. A group motivated to leave home may be trying to preserve a particular culture. So they may be a snapshot of some ancient culture, evolved into something new in isolation. They may have fused with a local people and have peculiar new ideas even if they speak a similar language. Or they may have preserved some long-lost blasphemy thought driven out of the world for good. And naturally they will have picked up new foods, diseases, and possibly enemies for your explorers to deal with.
Common evolution
China and Europe are two medieval setups in two distinct continental areas, separated by kilometers. Both discovered the same things nearly at the same time.
Their political and military evolution was similar, with barbarian tribes, empire lacking in control of its elements, concurrent small and powerful kingdoms.
Same thing for Japan.
You are in luck! We have an example and a counter example in history.
North Africa hosted a European style societies and is a mostly physically separate continent from Europe. Asia was mostly/ approximately the same tech level as Europe until about 1500.
Counter examples would be Southern Africa, and the Americas which didn't progress at the same rate as Europe because they didn't have access to the crops and domesticated animals that Europe had access to.
That is yes, if and only if the continent is connected to the trade networks (well enough to get crops and domesticated animals). If they are not then it will be whichever continent that has the most domesticated animals and to lesser degree most diverse portfolio of crops will win.
