First, I'll be that guy and say - don't backup your MFA key. If you lose your device, just jump through the steps of resetting it by contacting support.
While it doesn't necessarily defeat the purpose of increasing the security, and while it's also probably not likely that someone will attempt to steal your key, I don't think you're doing yourself any favors, security-wise.
But that's not what you're asking about.
When you say "on AWS it asks two codes to confirm, that makes me think it is device specific," I'm not sure I follow. Yes, it's device specific, in that you need the specific device that either scanned the QR code, or entered the key in, in order to auth via MFA.
But just because there are two fields, it doesn't mean that there are two different QR codes or MFA keys you need - you just need the one they show you.
After you set up your authenticator, you enter the first code you see into the first field, then wait for that to cycle out, then enter the next one into the second field. Asking for two codes just ensures that your authenticator is working correctly. It's not any different that other services that use an authenticator as MFA - some only ask for the first code that appears, some ask for two. (Personally I think two is better.)