why can I not sign in to my apple account with the Apple TV app on my MacBook air

The dialogue box closes and I’m back into the screen which invites “watch now, movies, ….. library “ if I click on library I’m directed to “sign in” and so start the whole process again.

Try signing in to the Apple TV app from the Account menu. (Use your Apple ID account, not the Mac login.)

A small aside: the Apple TV app logo on my MacBook is a circle not a rounded square. Any significance?

I think the Apple TV app icon is supposed to be circular on macOS Catalina 10.15: [Reference]. So that seems fine, just a slightly different icon design for that OS version.

If your issue persist, then please contact an expert from the Apple Support staff for further assistance. Contact Apple for support and service (phone numbers) or start your support request online (chat/​email/​callback).

What happens instead, when you try to sign in?

Do you get an error message? What does it say?