There is no "Apple Authenticator app". You can download the Microsoft Authenticator, Google Authenticator, or any number of other 3rd party authenticator apps. Apple doesn't make one. Apparently apple tech support is useless about their authenticator app as is the app store

While you can install third-party authenticator apps like Google’s or Microsoft’s code generator apps, you can also opt to use Apple’s Passwords app. Any of these apps is a good choice, as long as you are using multi-factor authentication. To use Apple's authenticator feature, you’ll need to set it up for each account individually.

Some websites will offer different authentication options, such as SMS. To use the Apple Authenticator app, tap Authenticator App and select Continue. Etsy offers an App Store link to Google Authenticator, but since we're using the Apple authenticator app, just tap Continue. The site will display a QR code, as well as a setup key.

Apple's built-in two-factor authenticator works seamlessly with Safari and apps on both iOS and macOS. The 2FA will automatically save security keys when setting up new two-factor settings on your accounts, and will auto-fill the generated codes when you log in to those accounts.