Upgrading to SQL Server 2012 from SQL Server 2008 I have upgraded my SQL Server 2008 R2 Enterprise edition to SQL Server 2012. Even after upgrade I can see &amp; use SQL Server 2008 R2 instance without any issue. Did my upgrade went fine? 

I can open default instance both from SQL Server 2008 and 2012. Also I have updated the compatibility level of all databases to 110.

After in-place upgrade



Edit - version

After running 

SELECT @@Version


SQL Server 2012

Microsoft SQL Server 2012 - 11.0.2100.60 (X64) 
    Feb 10 2012 19:39:15 
    Copyright (c) Microsoft Corporation
    Enterprise Edition (64-bit) on Windows NT 6.1 &lt;X64&gt; (Build 7601: Service Pack 1)


SQL Server 2008 R2

Microsoft SQL Server 2012 - 11.0.2100.60 (X64)   
    Feb 10 2012 19:39:15   
    Copyright (c) Microsoft Corporation  
    Enterprise Edition (64-bit) on Windows NT 6.1 &lt;X64&gt; (Build 7601: Service Pack 1)
Turning my comments to answer :

SSMS is just a tool to connect to SQL Server.

You did an inplace upgrade, but were connecting to SQL Server 2012 using SSMS 2008R2 and SSMS 2012.

Using T-SQL - will give you the correct version:

SELECT
   SERVERPROPERTY ('MachineName') AS PhysicalMachineName,
   SERVERPROPERTY ('ServerName') AS SQLServerName,
   SERVERPROPERTY ('Edition') AS ServerEdition,
   CAST(SUBSTRING(@@Version,charindex('SQL',@@version, 1),15) AS VARCHAR(255)) +  ' + ' 
       + CAST (SERVERPROPERTY ('productlevel')AS VARCHAR (50)) + ' + (Build'
       + CAST (SERVERPROPERTY ('ProductVersion') AS VARCHAR (50)) +')' AS ProductVersion,
    RIGHT(@@version, (Len(@@Version)-charindex('Windows',@@version, 1))+1) AS [O.S.],
 SERVERPROPERTY ('Collation') AS Collation


Best practice:


Before upgrade - run the above script to get the version and after upgrade, run it to make sure that you are on correct right version.
Always use the latest version of SSMS (regardless of SQL Server version).