For anyone who might see this error: “ERROR for site owner: Invalid site key”, with the Advanced Google reCAPTCHA plugin on WordPress, it helped me to double check to see if I had reCAPTCHA v3 activated in multiple places (example: Contact Form 7, WP Forms, Elementor, etc.).
I had reCAPTCHA v3 activated in Contact Form 7 and the Advanced Google reCAPTCHA plugin, and had to update the Keys in both plugins. The error was removed after that.