If you use an Android phone that's rooted, you can use Titanium Backup to copy your Google Authenticator app data between devices. It's up to you to copy that data somewhere else, but it is a very low level alternative to storing everything in 1Password or similar.
Even without rooting, just do that that next time when you register for 2FA, save that QR image, or screenshot it, and/or save that 16+ chars string somewhere safe, same as where you save passwords. Phone died/changed/lost? Install Google Auth, rescan that QR/Screenshot.
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