Can't authorize MacbookAir

You said that you couldn't find Manage Devices in the Music app, perhaps it should be Account Settings? You can see the authorized devices after clicking on it.

I also found an official guide which may be helpful to you. Here is the link: Authorize or deauthorize your computer for Apple ID purchases - Apple Support Maybe you can try reauthorizing your Mac to see if the problem can be solved.

"Can't authorize MacbookAir: Apple won't let me authorize this MacBookAir because it says I have more than five devices. When I look at my listed devices under my Apple ID, it just shows five, including this MacBook Air.[...]"

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5 Computer Limit:

You can only have 5 computer synchronized with your Apple ID. So, you'd have do sign out of iCloud on another computer.

Deauthorize All Option:

Try Deauthorize All. You can only Deauthorize All items once every 365 days. So, you may have to wait up to a year before you can perform this, if you have no access to any of these computers.  Go Here: Authorize or Deauthorize your Computer for iTunes Store purchases - Apple Support

"Thank you. What worked was an online chat with Apple support. They were able to deauthorize all, and I could then authorize the current devices."

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You are most welcome.