Private APIs and anti-tracking mechanisms are part of the operating system, not apps. As long as Apple's services don't get those same access, neither does any other app from a 3rd party app store.
Part of Apple's anti-tracking requirements are non-technical and exist only as App Store policy. That's the part that opponents of this change are concerned about losing should they end up pressured into using another store.
That said, there's not a whole lot an app can do to track you (without requesting special permission) that a random website you visit couldn't.
