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> If you lose a key, do you need a backup master password like using auth apps?

Yes. If you only have one key enrolled and no other recovery mechanism, you're now locked out forever. That's why Apple are pushing you to have two keys as a minimum.

When you log in to your account you have to provide your password and also use the key.

You can choose a key that has an additional factor built in, such as a fingerprint reader.

If you lose one of the keys you can authorize a new one.

If you lose both of the keys and all the Apple devices that you can log into directly, e.g. with fingerprint sensor on a Mac, then you're SOL.

Get a pouch for your water bottle that has a pocket for your keys.

Correct. Apple will prompt you to create a recovery key or assign a "Recovery contact" before you can enable support