Can I add my MacBook to iCloud

Do you mean backup your MacBook Air to iCloud? If so, no.

Apple offers iCloud Backup for iPhones and iPads but not for Mac computers.

It is possible to sync data from a Mac to iCloud if you turn on those options (e.g., iCloud Photos, iCloud Drive including the Desktop & Documents option). Syncing has some of the benefits of a backup (and the benefit of being offsite) but is not the same as a backup. Major differences include the fact that not everything on your Mac can be synced and the fact that if you delete something, accidentally or otherwise, there is only a limited time period (30 days in most cases) during which it can be recovered.

Also see the Apple article: How to back up your Mac - Official Apple Support

And more information on syncing in:

Set up and use iCloud Photos - Apple Support

and

Access and manage your files in iCloud Drive - Apple Support

and

Add your Desktop and Documents files to iCloud Drive - Apple Support

You can have multiple Photos library on a Mac but only one (the System Photo Library) can be synced with iCloud. See information in Designate a System Photo Library in Photos - Apple Support

You can use Time Machine or 3rd party software to backup a Mac to an external drive. One example is Carbon Copy Cloner https://bombich.com/

However, if you have imported your photos into the Photos app, even though you can back them up with 3rd party software, you should use the Photos app to access them. It is technically possible to access them from Finder but they are not designed to be accessed that way and doing so can cause serious problems.

Thank you for your post! This is the first time that I have understood that iCloud is a backup only for iOS devices and simply provides limited syncing for Macs (desktop, documents and iPhotos library).

I use a different library for my 60k MacBook photos so that I don't clog up my iPhone and iPad(s). Am I correct that Apple has no cloud based backup services available, for that?

Are my only options physical ones like Time Machine? And if I use Photos how can I back them up physically outside of Time Machine without them requiring Mac software?

Of course I am concerned about relying just on physical copies. Thank you!