I'm not certain about this, but I suspect it's works the following way.
The website sees that you are accessing it with mobile Safari (browsers do identify themselves this way) and generates the code so you get the popup reminder. The fact that you've been reminded is stored in the cookie for the site.
Now, if you are in private browsing mode (no cookies stored), set to reject all cookies, or your cookies (data) gtt cleared, the popup is going to happen again, and again, and again. You'll also get it again whenever you use another browser, when it offers a different (but still iPad specific) browser id.
It is the site that generates the popup, not the iPad (though it uses a local reminder window). So it is the site that needs to track whether you've been reminded.
Perhaps a checkbox in the site's options could be created to permanently disable the reminder? Hard to say whether this would be easy or difficult for the developers.
Now, as I said, I could be completely wrong about how this works, but given the intermittent way I receive this popup, I'm betting I'm pretty close.