I can only comment on the SQL Server concept of databases, so this may or may not apply to other RDBMSs.
In SQL Server, a database is (1) the boundary for HADR (high availability and disaster recovery) and (2) the boundary for security.
Performance? SQL Server shares one buffer cache and one log cache for all sessions for all databases so there is little performance implication of one database vs many databases. If you have 15 megabytes of changes, that means 15 megabytes of traffic to be written to disk, regardless of how many databases it belongs to. Of course, more files does add a very small overhead in management traffic.
Connections? One client application makes one connection to the SQL Server database engine service and can then query multiple databases. The client does not need multiple connections for multiple databases. However, since a database is the security boundary then if you want a session to query multiple databases then the login will require permissions set separately in those databases.
In summary, usually one database is created for one application. For example, the Payroll database for the Payroll application.