Most services that support it seem to have a standalone 2FA authenticator app (Microsoft Authenticator, Google Authenticator) instead of integrated authentication in their mobile apps (admittedly, these services are far larger than one-app products, so a single-port-of-call authenticator app makes sense from this regard).
If I want to make a system that can be accessed via a website and via an app, what are the security differences between making the app its own authenticator (and the authenticator for website logins) versus having a separate standalone authenticator app?