SSPI handshake failed with error code 0x8009030c - only stops when SSMS is closed
Possibly the developer is using a 3rd-party tool that provides a plug-in for SQL Server Management Studio. The 3rd party plug-in is started with SQL Server Management Studio and is trying to connect to the SQL Server using the credentials the process was initially started with. 

If the initial process owner of the 3rd-party tool is running with an un-privileged account that does not have access to he SQL Server instance the developer is connecting to, then you will possibly see multiple error messages in the ERRORLOG file and in the Windows Event Log on the target server (see Link below for examples).

If the initial process owner of the 3rd-party tool is not allowed to delegate permissions on its own object, then you might see the SSPI Handshake error.

There is a Microsoft Blog article titled Common ‘SSPI handshake failed’ errors and troubleshooting which provides you with some ideas on how to find the main cause.

Most of the SSPI Handshake erros are because a process can not handover/delegate the permissions during the SSPI Handshake to another user/process.