Isn’t the value of two factor auth that it requires a physical device (your phone or computer) with the auth key to authenticate you? Then why don’t many two factor auth apps seem to support syncing? If it’s fine to do so, are there any open source cross platform apps that sync keys?
It’s considered bad form to do what you’re asking but most 2fa apps have a backup restore scheme now. Is that enough?
A physical token only authenticates itself as “something you have” if there’s no way to extract the key from it. In practice non-hardcore deployments usually have a backup procedure but in principle, if you want multiple tokens, they should have separate keys. What you’re asking in simplest form involves storing the key on a server where it can potentially spill in a server breach or the like. If the key protects something very valuable, that can be dangerous. If it’s for your old Reddit account, you might decide to do it anyway.