I don't think there's an online backup it does have an export option but you can only export them if you are going to move the authenticator to a new device because it would generate a QR code that you need to scan to another device.
It has an online backup. Some people said it was optional but I updated the authenticator and I saw that it has been backup online. I did not know how it happened until it was too late.
I never tried exporting my authenticator with email I saw this as another option but you can just ignore it and use the QR code option instead if you want to export and move it to a new device.
I do not believe in QR code backups. I backup the secret code generated on the site which should be backed up. I prefer it that way.