One dayway to accomplish this would be to use the host_process_id
column from sys.dm_exec_sessions
This host_process_id in this table maps to the PID in task manager.
In this case we could run the following query to get our host_process_id, and then take the result and check in task manager which application currently has a PID matching the result.
SELECT session_id, host_process_id FROM sys.dm_exec_sessions WHERE session_id = XXX
You can then check which account is running that process.