Would it be possible to migrate my database to SQL Server Std?
Yes, a backup and restore approach would work for this situation. For a detailed migration steps (ignoring the ones that you dont need - as it is for migrating from lower to higher version), refer to this answer.
Can I install my SQL Server 2012 Std on the same server I have of my SQL Server Express?
Yes, Express and standard editions can co-exists together. Make sure that your server have enough memory, CPU and disk space. Also, make sure to cap the Max memory setting on the instances appropriately.
Is there any risk doing that in terms of sharepoint compatibility?
We have SharePoint running in our environment on SQL server 2012 standard edition, so I dont see any issues. What risks are you referring to ?
NOTE: I have installed SQL Server 2012 Std on same server but the db size limit still persist, is this normal?
No that is not normal. See Database size Maximum Capacity Specifications for SQL Server 2012.
Below script will give you more detailed info about the version you are running :
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