Q:

How to assign private ssh key to only one user? 

I have a ssh-key authorization on my server.  I've added public key for two users: user1 and user2 to authorized_keys.  How can I restrict to login by some user only with his key?  User1 should only can login to server with his private key (he shouldn't can login as User2 with his private key - now that's working).  Why it's working? 
Example:
User 1 logging as user1@server with his private key - login successful. 
User 2 logging as user2@server with his private key - login successful. 
User 1 logging as user2@server with his private key - login successful (but shouldn't!). 

A:

Each user should have its own authorized_keys files.  By adding the public keys of two users to a single authorized_keys file (which sounds like what you have done) you are granting access for 2 users to the same account.  The public key for user1 should be in the authorized_keys file associated with user1, and the public key for user2 in the authorized_keys associated with user2.  
You do not state the OS of your server, but if it's Linux the public key for user1 should go in "/home/user1/.ssh/authorized_keys", and the public key for user2 in "/home/user2/.ssh/authorized_keys".