I want to edit photos in LR on my laptop and desktop PC with the folders, files and catalog on one external USB 1TB SSD, plugging it into the appropriate computer: laptop when I'm away, desktop when I'm home, i.e. never at the same time.
I can import photo files into LR on my laptop, creating a folder and editing them OK, then saving a catalog backup on exit: all to the usb ssd.
Then when I connect the usb ssd to my desktop PC the folder created in LR is not visible in Windows File Explorer and LR fails to launch as the catalog is corrupt.
Using File Explorer/Properties/Tools/Error Checking - there are errors on the SSD but Windows 10 can't repair them.
You need to find out what the errors are on the SSD. For instance, is it properly formatted to function on a Windows computer? Another thing is that you don't want a backup catalog because it is going to be a zip file. If you're going to use the same catalog then perhaps you should just move the working catalog to the SSD in the first place Along with all the images and use that catalog on either computer. You don't have to do it that way, just a suggestion.
But your first order of business is to figure out what the errors are on the SSD. That isn't a Lightroom problem.