-- To find out which database you are connected to:
SQL> select name from v$database;

NAME
---------
DBSTAR

or you can query the v$parameter view:

SELECT UPPER(value) db_name
  FROM V$Parameter
 WHERE UPPER(name) = 'DB_NAME';

DB_NAME
--------------------
DBSTAR


You can also tailor the SQL prompt to identify the user connected to the database and the database itself.
Add this to your login.sql to have the user id you connected as, as part of your sql prompt:

-- Store username 
col xuser_name new_value xxuser_name 
select username xuser_name from user_users 
  where username = user; 

-- Store database name 
-- Add this to your login.sql to have the database you are connected to as part of your sql prompt:
-- You need select privileges on v$database for this to work
column xsid new_value osid
select name xsid from v$database;
--
set sqlprompt '&osid:&xxuser_name> '