why can't i sign into apple tv on my MacBook air

On macOS 10.14 or earlier, there is no Apple TV app. That app is included with macOS 10.15 or later.

On macOS 10.14 or earlier, instead access Apple TV+ content on the web, at tv.apple.com.

Apple doesn’t make smart TVs, but Apple does make an external streaming device for TVs: the Apple TV box.

Additionally, select smart TVs and select streaming devices can install the Apple TV app. Apple TV app - Devices - Apple

These allow users to access Apple TV+ (subscribe), iTunes Store movies (buy, rent) and TV shows (buy), and select premium Apple TV app Channels (subscribe).

Apple TV+ is a subscription service for users that sign up for that. No-one gets it automatically. In combination with certain purchases, an extended free trial may be available to new subscribers. E.g.: [Apple’s offer] [T-Mobile offer] [PlayStation 5 offer] [LG offer]

Please contact Apple Support to inquire about compatibility for signing in to tv.apple.com with Safari on OS X El Capitan 10.11. Contact Apple for support and service (phone numbers) or start your support request online (chat/​email/​callback).

Apple says that “customers can also subscribe to and watch Apple TV+ at tv.apple.com in Safari, Chrome and Firefox”, but doesn’t mention browser versions or OS versions. [1] [2]