It seemed odd to me that a Web site could write to or read from the clipboard without the user approving it. That would be a pretty obvious security and privacy issue. From what I gather, on Chrome sites can write to the clipboard without approval, but they need approval to read. On Firefox and others any access requires permission. Thus this exploit seems limited to Chrome users.
@SkaveRat pointed out that it doesn’t require permission, only interaction. So likely there’s a button that’s clicked that writes to the clipboard, and most browsers are susceptible to this.