Switching phones means transferring a lot of things.
Most of it is handled by sync. What it logically can't manage is moving the data in Google Authenticator. I use it wherever possible. But each service handles the transfer differently. Google first verifies it's really you, then offers to transfer to the new device. For MojeID you need to disable OTP (confirming with one of the backup codes) and set it up again. Dropbox verifies you the same way as Google and then offers to re-configure. Your Microsoft account requires disabling and re-enabling. If you use Authenticator for WordPress you just scan the QR code again. I expected far more trouble and complications — everything went far more smoothly than expected. Nothing to be afraid of.
In 2018 I switched to Authy