How to get user name to appear in Alerts
Alerts generated by the SQL Server Agent don't include this information, and you can't add it, unfortunately. 

One option would be to set up an Extended Events session to capture the error_reported event and filter by the specific error number. This will allow you to capture additional info such as program name, username, client machine etc.

Log this to a file and use an Agent job to periodically query the file and alert if entries are found.

Links:

Login Failures in Extended Events: https://www.mssqltips.com/sqlservertip/3214/use-extended-events-to-get-more-information-about-failed-sql-server-login-attempts/

Extended Events Quickstart Guide: https://docs.microsoft.com/en-us/sql/relational-databases/extended-events/quick-start-extended-events-in-sql-server?view=sql-server-2017