you can rename the database , but a fully qualified database is known by instancename/database name anyway.

select convert(varchar(128),serverproperty('instancename'))

select @@servername

select convert(varchar(128),serverproperty('machinename'))

e.g.

instance102

myserver\instance102

myserver

add to the database name as required e.g.

select convert(varchar(128),serverproperty('instancename'))+'_'+db_name()