The REGISTER DATABASE command registers the target database in the recovery catalog tables. After executing REGISTER DATABASE you will see your target database in the RC_DATABASE view.
Issue
Note that after dropping the pre-created catalog and recreating a new catalog in the RCAT tablespace, I received the following error when attempting to register my database:
RMAN> register database;

RMAN-00571: =======================================================
RMAN-00569: =============== ERROR MESSAGE STACK FOLLOWS
RMAN-00571: ===========================================
RMAN-03009: failure of register command on default channel at
ORA-04062: timestamp of package "RMAN.DBMS_RCVCAT" has been cha

Exiting RMAN and reentering solved the problem.