Inability to use my Lightroom 5 software

@JP Hess is correct about Lightroom5 being unsupported for many years. But having said that, there's no reason why it won't continue to work reasonably well on Windows systems....I can run it OK on my Win10 system, and I even used LR3 yesterday when checking something out in another forum. I'm guessing that you used the original DVD to install LR5 on the new computer, which would likely have been the base version LR5.0. So when you launch that version it should indeed tell you that an update is available (i.e. to LR5.7.1, which was that last update issued for LR5). The problem is that the download links for all the LR5 and LR6 updates were removed by Adobe some years ago, so you need to find a link if you can. Try this: Lightroom 5 for Windows (957 MB)

What you actually purchased was a license to use an installation of software that is owned by Adobe. And if my memory serves me correctly, that license was for a specific period of time. Lightroom 5 is no longer supported nor updated by Adobe. It is notably obsolete, it hasn't been modified for years. Operating systems have changed and that has necessitated the modification of software to function properly with today's operating systems. Lightroom 5 might be sufficient for YOUR photographic needs, but everything else that you use to support your photographic needs has changed. Whether or not Lightroom 5 will function for you is not the responsibility of Adobe. The company has moved on to newer versions that meet the requirements of today's operating systems. If Lightroom 5 works for you then that is great. If it doesn't then it's up to you to figure out how to make it work or to decide on some other solution to meet your photographic needs.