I just tried til help a friend with installing Lightroom on Mac OS Ventura. Creative Cloud was installed, and I used the button Install on Lightroom in the Creative Cloud app. The installation was performed, and reported Completed. Lightroom button changed to Open. Clicked on Open, but nothing happened. Wide search in Finder for the Lightroom app file, but it was not found on the entire disk. Edit: As Lightroom is tested compatible with Ventura, this is probably related to the installer app or lack of priveleges. See my comment below.
This morning I made a second try to install Lightroom 6.0 cloudbased on Ventura, but first I granted write permission to the /tmp/ folder. I also checked the installation folder in the CC-app. That was correct, and other applications in the CC-app, like Bridge, was installed correctly. The application is compatible, but will not install by running installer from inside or outside of the CC-app. This affects new customers and users the try reinstalling due to some issue.