I recently needed to reinstall macOS via USB stick on a laptop that has not been used for a while. I was able to boot from the USB, but when running the installer I was greeted with the “This copy of the Install macOS Catalina application is damaged…” message. Curiously, the same USB stick worked fine on another Macbook.
During the time in storage the laptop battery ran out of power completely and the system clock was reset to Jan 1 2019. Using the “Utilities – Terminal” window I was able to set it to today’s date with the “date” command.
It is probably related to a signature on the Installer image that is only valid from a certain starting date.