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?
Bitwarden has a FOSS client app and FOSS server apps exist (though the default service is not FOSS).
Syncing 2FA keys brings the danger with it that you accidentally sync the key to the device that is used for the first factor thus making it not 2FA anymore.
the default service is not FOSS
You mean server? If so, the server is also open source (https://github.com/bitwarden/server), but the default instance (bitwarden.com) is not totaly free - you have to have a payed subscription for some of the features. If you self-host, then you have all the features (free and/or premium) - and this can also be done with Vaultwarden which is a FOSS alternative to the official server.
