There is a PHP application that uses only one MySQL username. And there are several databases in the MySQL server. Each database is used by a different company. Should I keep "one MySQL" for the entire system or should I switch to "user per database"? Will I have performance increase if I switch to "user per database"?
Using different users will not reduce or increase the performance. It is clearly a security issue.
Usually you don't want users from company A to be able to access company B data.
I would recommend to create separate users for each application and grant them only the necessary permissions.
If you run the application using a user with admin rights you risk a lot when the application gets compromised.