No, because this isn't a common occurrence or part of any best practice.
The user you're referring to may be called
sa, or something similar. In many cases these users are disabled or not used frequently to increase system security, though other users with similar privileges most likely exist to take their place.
In most normal cases, naming a database or database server instance the same as the default superuser might be unclear, confusing, and provide no actual information about the data or its use.
Database names may commonly indicate the name of the application it supports, include an indicator of the environment it is in (dev/qa/prod), and/or the type it is (oltp/olap).
For example, some of Microsoft's sample databases for SQL Server are named "AdventureWorks" and "AdventureWorksDW", both named after the fictional company the databases are built around. The second one includes "DW" to indicate it is a data warehouse and not an OLTP database.