Many web services require a database and quite often that is MySQL. Typically, the installation instruction of the software tells you to create a database and a user with permission for that database.
Is it OK to name the user and the DB the same?
For example, I am installing Openfire right now and I created a MySQL database "openfire" and MySQL user "openfire". Should I expect any problems? What is the convention?
Yes, that is absolutely fine and quite a common practice. They are mutually exclusive in use and will not cause any problems.
All you need to ensure is that they comply with username and schema naming conventions:
It is ok to name the user and the DB the same, you should not have problem.
But from a security point of view you should maybe use another username.