What is the purpose of keeping filegroup or filestream backup in MSSQL?
So I tried to restore only filegroup of filestream to other database, but I get an error message


As Laughing Vergil originally mentioned in a comment, your question is answered in this great Stack Overflow post: Restore filegroup to different database

The gist of the answer is that you always have to start by restoring the database with the PRIMARY filegroup.  Then you can restore additional filegroups (like the filestream one) after that.  This called a "piecemeal restore" and is described in the documentation.


  Is it possible to restore only filestream to other database 


No, as mentioned above it needs to be restored along with the PRIMARY filegroup.


  or restore only an old version of filestream to the current database of FILESTREAM?


Tibor responded to this part in a comment.  The basic answer is "no."  This is because of the way database recovery works - you can't restore different parts of the database that are from different points in time.

As a side note, there is a restriction specifically related to filestream filegroups:


  If a partial restore sequence excludes any FILESTREAM filegroup, point-in-time restore is not supported.