Hi! Сайн уу
So my understanding of the Apple Media Services Terms and Conditions is that because you are from Mongolia, your Apple ID only would be valid in the Mongolia iTunes Store. Your ability to download a particular artist wouldn't relate to whether that artist was ethnic-Mongolian or Mongolian-based, but rather whether the artist and/or their record label had licensed the rights to Apple for sale in the Mongolian Store.
For example, I'm in the US but was able to listen to The Vendebo, including their newest album "Bayarlalaa." For what it's worth, Chinii Nudeer (feat. Anir) is an excellent track, too! :-) The Apple Music Store in the US must have the rights.
If you have moved from Mongolia and need to change your Apple ID country or region, you can. But it wasn't clear from your post that this was your issue.
If you aren't able to download music that you think you should be able to, you should contact Apple Support.
Good luck!
JDee