Which component of SQL Server architecture handles the user login requests?
There is not a dedicated "component of the SQL Server architecture" for this.  The protocol layer, including the whole login process is handled by the Database Engine.  The SQL Server Database Engine is a single large process that hosts


  "the protocol layer, the query processor (also called the relational
  engine), the storage engine and the SQLOS."


--Microsoft SQL Server 2012 Internals, pp 10 Kalen Delaney, et al

This is in contrast to Oracle, which has the "Oracle Net Listener" which is a separate process. https://docs.oracle.com/cd/B28359_01/network.111/b28316/architecture.htm#NETAG209