Personal accounts in Microsoft Authenticator, such as your bank, Amazon or a personal Gmail, store a six-digit code generator. That generator is included in the app's backup and comes back on the new phone ready to use.
Work or school accounts are different on purpose. The credential that approves your sign-in never leaves the phone it was created on. The backup restores only the account name as a placeholder, and the new phone has to sign in once to earn its own credential. Passkeys stored in the app are tied to the phone too, so they are created again on the new one. This is why people who restore a backup and see "Sign in to add your account" in red think something went wrong. Nothing did.