There isn't any "show databases" equivalent in Oracle.
FWIW, I use Postgresql and Oracle, not Mysql, and am guessing that Mysql is similar to Postgresql in this respect. Postgresql supports the "database cluster" part of the SQL standard so you get one cluster per instance with one or more databases per cluster. Oracle does not support the "database cluster" part of the SQL standard so you only get one database per Oracle instance.
To determine the name of the currently connected database, you should be able to:
select global_name from global_name;