They are listed on the dashboard, you point and click, select the version you want and the database is ready and running. There is no requirement on your part to handle the installation.

Would you call it a managed database, i.e. does it come with continuous backups and things like that?

Nope not managed. Only the Shared MongoDB database is managed. Does not provide continuous backups although planned only snapshot backups at the moment. We intend to offer a managed shared offering for the other databases as well but not a priority as MongoDB was and still is the most requested database.