How can we identify AlwaysOn Process?
Is there any way we can identify it, so that we can monitor/restrict how much resources being/will be used by AlwaysOn through resource governor.


I would absolutely not advise trying to do something like this. The point of Always On (I'm assuming you mean availability groups, here) is generally for high availability, disaster recovery, or multiple copies for read heavy workloads without using Replication. 

Always On is accomplished through system processes on hidden schedulers (depending on version of SQL Server) and there wouldn't be a straight-forward way to do this with resource governor.


  I'm able to get application name for SSRS, web App and SSIS using APP_NAME() function, but not able to get application name for AlwaysOn processes.


Always On Availability Groups aren't an application but parts of the system. They use all of the normal built in mechanisms to SQL Server with only a few specifically built for it. 

For example, part of the process is to write log information to disk. This is no different than writing any other log to disk, when you look at it from the outside. If you were to govern the amount of IOPs that could be done to a database then it would affect the entire database, not just the Availability Group process.