Using various copy methods, no content is made available via the Clipboard (Finder → Edit → Show Clipboard) when using add-ons like Studio Code Server, ESPHome, or Advanced SSH & Web Terminal. This works correctly when using regular web browsers (Safari, Google Chrome) on the same computer.
One of the add-ons produced this warning: Unable to read from the browser’s clipboard. Please make sure you have granted access for this website to read from the clipboard.
Unlike iOS, the macOS System Settings app does not have a container for Apps. One can add apps to certain settings but I haven’t found a way to change permission on the clipboard.
It is not the clipboard you should change permission on, but the browser.
Apple could be implementing it in their browser also, if you are using Safari.
Safari and Firefox is the two mainstream browsers that fight for first place in being private, so I would not be surprised.