I've read that CPanel has a function which allows you to backup a MySQL database to remote FTP site of your choice. I went to the backup page in the CPanel control panel (on a shared server), and don't see any FTP option there.

Is it still possible to backup a MySQL database to a remote FTP site daily?

If you are an end-user of cPanel (as in you do not have access to the WHM interface) then you cannot do this... at least through cPanel.

You seem to be describing the feature in WHM where an admin can have an entire server backed up and the backup sent off-site automatically using FTP.

I know when I was in charge of a large website on a cPanel-powered server, what I would do is set up a daily cron job to do a sql dump to a file. Of course, going beyond that with automating sending backups off-site is a good plan in case of catastrophic hardware failure.