No, legacy/perpetual versions of Photoshop were not designed or tested to work on macOS 10.15 (Catalina). They are not supported in any way for use on macOS Catalina.
Adobe does not recommend that customers using old versions of Photoshop upgrade to macOS Catalina.
Incompatible versions (version 19.x and earlier) will not display in the ‘Older Versions’ list within the Creative Cloud desktop app.
Older versions use 32-bit licensing components and installers. Therefore, they cannot be installed and activated after upgrading to macOS Catalina. Upgrading to macOS Catalina with an older version already installed on your computer may allow the application to function in some capacity; however, you will not be able to reinstall or activate the application after the macOS upgrade.
Before upgrading to macOS Catalina, you may uninstall the older versions as the uninstaller will not work after upgrading. If you have already upgraded to macOS Catalina, you can use the Creative Cloud Cleaner Tool to uninstall older versions.
You will have to downgrade from Catalina if you wish to install and use the earlier perpetual versions.