I am not an expert on this at all, but I think being able to sync 2 factor authentication data with a 3rd party provider and access it through the web would defeat the purpose of 2FA.
But maybe I am misinterpreting what you mean ^^"
Sorry if you already know this, but since you mentioned the issue of losing your device, I thought maybe you didn't ^^"
If you do lose or otherwise make your 2FA device unavailable there are usually security measures to get back into a good state. It could be through a unique recovery key, you usually can get a copy of from the place you enabled 2FA, but it can also be some secondary device being sent a recovery code. Some services will let you access the recovery keys several times, others will only allow you to copy it during initial setup.