Notifications from Apple on MacBook air

I've had multiple Mac computers and iOS devices and when I get a new device I get a pop up similar to what you mention (it just appears on the screen, as shown in the image below -- so I don't have to "do" anything). I have 2-factor authentication enabled, and you may check if you do as well. This site has more details.

You write that you are asked to "enter a password for my macbook AIR". That may be true, but also be careful to distinguish between the verification code they send and your MacBook Air login -- two very different things, and if you confuse one for the other I could imagine the frustration.

You may also find this page helpful as it specifically deals with Apple ID sign-in on different types of devices and (at the very bottom of the page) gives links to help in the situation you describe regarding forgetting your Apple ID.