This is only really a problem because on my mobile phone I have the JRiver App that I use as the remote and this app sees only one library.
Renaming the Library won't fix this, although that would be a good start.
As you have Restored a Library Backup from one machine on to another, both are now using the same Access Key for the Library Server. So JRemote (I assume you are referring to JRemote, and not Gizmo or something else) can only see one MC Server.
Go to "Options > Media Network" on the second machine and note your Access Key. Then turn off Media Network by unchecking the "Use Media Network to share this library and enable DLNA" setting, and then turn it back on. You should get a new Access Key, I think. Access Keys are reused when possible, but renaming the Library may force MC to get a new Access Key.
If you don't get a new Access Key, you may need to shut down MC on machine 2, wait a minute, then restart it and try again. Or maybe even shut down MC on machine 1 and 2, start MC on machine 1 so that it reports its current IP Address, which will take a few minutes, and then start MC on machine 2 again and turn on Media Network to get a new Access Key.
However you achieve it, you need a different Access Key on machine 2.