Hey folks... I had a similar problem because I'm running 2 different websites (with 2 different domains) based on WordPress... and I installed this same plugin on both sites. But after some tooling around, I got it resolved.
This explanation might be a bit complex, but hang with me. In the end, it's actually pretty awesome. If you want to skip the explanation, just skip down to "Solution."
For some reason, simply inputting an API key isn't enough -- you also need to be logged into wordpress.com every time you view your stats. (Yes, in addition to being logged into your own blog's wp-admin.)
The problem is that, in order to have multiple API keys, you need to have multiple wordpress.com accounts. And when you log in to wordpress.com with one account, it overwrites any cookies that had you logged into any other wordpress.com account. In other words, if you login to wordpress.com with "Account A" you can see the stats for "Site A" but once you login with "Account B" you won't be able to see the "Site A" stats anymore. And you'll get that pesky error message you complained about.
OK, so here's what you need to do. Logout of wordpress.com (you may need to open a new tab/window since the "logout" link doesn't always appear in the pages that are embedded in your own dashboard).
Now, go to the Blog Stats page for one of your blogs.
After you login to wordpress.com there (with the account linked to that particular site), scroll down to the bottom and you should see a place to add users. Add the e-mail address(es) linked to your other wordpress.com account(s).
Logout of wordpress.com and repeat the process for your other blog(s).
Once you're done linking all the sites to each other... you can login to ANY one site, and see the stats for ALL of them. There will be a pull-down menu near the top-right of the Blog Stats page which allows you to switch around between your different sites. Cool, huh?
Disclaimer: I'm running both of my sites on my own hosting account. If you're running blogs hosted by WordPress on their own servers, I'm not sure if there's other factors which may render this solution useless.