TOTP/HOTP codes are defined by an algorithm (sha1/md5/...), secret (A826EF8...), and number of digits (I usually see 6 digit codes). TOTP additionally takes time period (30 second codes) and time (current time which maps to some 30 second block) as a parameter and HOTP takes a counter as a parameter. All of these parameters go into the function to generate the numbers as a result.

If you have ever set one of these up with a QR code, that QR scans to something like: otpauth://totp/ACME%20Co:john.doe@email.com?secret=HXDMVJECJJWSRB3HWIZR4IFUGFTMXBOZ&issuer=ACME%20Co&algorithm=SHA1&digits=6&period=30 (From: https://github.com/google/google-authenticator/wiki/Key-Uri-...). Notice all parameters I mentioned above are present, as well as a user friendly account name.

So to directly answer your question: a backup would in some way contain all the parameters above, possibly in that otpauth:// format, but could be json or something else.

I would not consider Authy to be a trustworthy backup. I assume they are storing these secrets for you and transferring them to other computers at your request. If you can't see the secret, you can't switch to a different app. (Take this last paragraph with a grain of salt, I don't know much about authy but it sounds like trouble. I use FreeOTP and other open source OTP apps).