Anyone know if you can use the authenticator
app on more than one device?
The normal way to do phone-based 2FA is a QR code with data of the format "otpauth://totp/yourusername?secret=1F56D7AFLONGBASE64&issuer=Amazon" where the secret is the secret needed for TOTP [2] one-time code generation.
As such, you can write down the secret (or print out the QR code) and scan it into other phones (or use it with tools like oathtool on linux) and they'll then generate identical codes to your main phone.
Obviously, if you store your TOTP secret alongside your password or keep a copy somewhere that isn't safe, there's no point in using 2FA. And if people fuck this up too often 2FA users will start insisting we install twenty shit proprietary apps (one for steam, one for salesforce, one for symantec vip access....) and nobody wants that. So use your new powers with care!
[1] https://www.eff.org/files/styles/large/public/2016/12/19/ama... [2] https://en.wikipedia.org/wiki/Time-based_One-time_Password_A...
Then again, it sounds like it's really easy to disable 2FA with just a simple phone call, so...
https://www.amazon.com/gp/help/customer/display.html?nodeId=...
oathtool --totp -b "key value"
where your "key value" is your secret (same thing you would get if you scan QR code). And then you just need to keep the secret safe, and you can run it on as many devices you need.
EDIT: just realized that michaelt had much more substantial comment.