Creating Linked Server Object to same instance
I've already worked it out...
The problem was incorrectly setting the provider string to SQLNCLI.
When I removed it (leaving it empty) it connected immediately...

Linked Server: NewServer
Server Type: Other data source
Provider: Microsoft OLE DB Provider for SQL Server
Product name: {empty}
Data source: LOCALHOST\SQLEXPRESS
Provider string: {empty} (UPDATED)
Security, Login: Using the login's current security context