A Microsoft app for iOS and Android devices that enables authentication with two-factor verification, phone sign-in, and code generation
It sounds like you're stuck in an MFA loop because Authenticator is asking you to approve the sign-in on the same phone where you're trying to sign back in.
On the sign-in screen, look for “Sign in another way” or “Other ways to sign in.” If you previously added SMS, phone call, security key, or another verification method, you may be able to use that instead. Microsoft recommends this when the primary verification method isn't available.
If this is a work or school account and there is no other verification option, you'll probably need to contact your Microsoft 365/Entra admin and ask them to reset your Authenticator/MFA registration. After that, you can register the phone again.
I wouldn't keep reinstalling Authenticator, since the problem is more likely the account's MFA registration than the app itself.