Sidenote: I merged now two other threads into this one, as they all requested the same. Therefore I reframed the title here, as currently, the available backup option for the authenticator app is “bound” to the device’s backup system (i.e. “protected with phone backup”, as stated in the corresponding blog article), which is either iCloud or Google One.
So all three threads requested more backup options and therefore also independent one’s from the “phone backup method”.
(screenshot from the Authenticator FAQs)
Here is a somewhat related but different request to this one (but regarding offline exports): Encrypted Offline Backup/Export option