First of all, the secret key in discuss is defined as when you cannot scan a QR code in a 2FA setup process, there is a code that you can manually enter into your TOTP-generating device/app.
Some services provide one-time use backup codes for two-factor authentication, for the sake if you lose your code generating device, but some don't provide backup codes. For those services who don't provide backup codes, I tend to store their secret key.
My question is, is it safe to store those secret keys the same way one store backup codes? Is there any differences?
I believe the secret key displayed on screen for user to manually enter are already included in the QR code, and it can be used to entered into any 2FA apps, to make different apps on different devices to generate the same TOTP codes. Please correct me if I am wrong about this.