Well, each country sort of has its own diverse culture, so there isn't exactly a country that will have the same culture as Mongolia, although some other Asian countries may have similar customs, but actually, Mongolia is quite diverse itself…

Perhaps you could take the following approach: which other cultures have influenced Mongolia, and which cultures have been influenced by Mongolia?

The map of the Khan empire, extending all the way to the Black Sea, is impressive. It was important for the trading routes and cultural interchange between Asia and Europe, for example.
For example, during the time of the great Khan empires, Mongolia imported Buddhism from Tibet.

Well, Kazak culture is not very far off. There is also Inner Mongolia, which is a state of China but is essentially Mongolian in culture.

Mongols practice mainly Tibetain Buddhism, so Tibet may not be far off.