You probably should. Your usage of the App Store logo is likely in contradiction with Apple's rules, and since it's their IP, they get to dictate how it's used.
The logo he's using is not the AppStore logo. It's the logo for a generic app with no custom icon on macOS. That said, apple still own that logo, and might have an issue with it.
It's literally the app store logo minus the circle. But you're right, the same icon is also used as part of the generic app placeholder icon. In either case, it does still belong to Apple.
Edit: In fact, thinking about it some more, it's very clearly much closer to the App Store logo than anything else. It's literally the white silhouette design from the logo (as opposed to the colored non-sillhouete version from generic app icons), placed on a blue button that's similar to the app store logo's blue background.
