If the two iPhones are different models it’s likely because the iOS version is different. Since the Apple TV app is built into the OS and not distributed universally through the App Store the Apple TV apps will be different between iOS versions.
For example the Apple TV app on an iPhone 6 or earlier is going to look different from an iPhone 13. This is because the iPhone 6 has a maximum iOS of 12 while the iPhone 13 is capable of receiving the latest iOS versions being released by Apple today.
Some Apple TV app update history:
iOS 12.3 This update brought with it a new redesign, Apple TV channels, Movie purchasing and rentals, and a Kids section.
iOS 14.3 Apple updated the Apple TV app again to introduce the Apple TV+ tab.
iOS 15.2 added the Store tab to the Apple TV app to let you browse, buy, and rent movies and TV Shows in the app.
iOS 12 Apple TV app icon
iOS 16 Apple TV app icon
iOS 12 Apple TV app layout
iOS 16 Apple TV app layout
