How do you create a new database in SQL Developer?
The term database has a different meaning in Oracle. The definition of database by Oracle:


  A set of files, located on disk, that store data. Because a database
  instance and a database are so closely connected, the term Oracle
  database is often used to refer to both instance and database.


Oracle Database Architecture

If you wanted to seperate data per application, you could do that by using seperate tablespaces and schemas.

Creating such a database is a complex process, and it is done with DBCA or SQL*Plus on the database server, not with a remote connection:

Creating and Configuring an Oracle Database

In 12c, the multitenant architechture was introduced, that is similar to the concept in SQL Server.

Overview of Managing a Multitenant Environment

If you have a container database, there you can easily create new pluggable databases from SQL Developer, when using a DBA connection.

Managing Pluggable Databases (PDBs) using Oracle SQL Developer