Can Claude for Chrome access profiles other than the one the extension is installed in?

I have a separate Chrome profile that I use exclusively for Claude for Chrome to be safe. When Claude for Chrome is active it brings up the 'Claude started debugging this browser' message, but I've noticed that the message appears in every chrome window that's open, even different profiles that it isn't installed in. Does this mean that Claude for Chrome has potential access to all open chrome windows, even those in different profiles? Even if it's not actively using them would that mean it couldn't potentially access those profiles if it's jailbroken? Haven't been able to find a concrete answer to this.