[wackget]

Stack Exchange sites have a horrible authentication system so using them as an example is a bad start.

However they could solve this "problem" in a number of ways, the most straightforward being to use subdomains instead of individual domains.

I put "problem" in quotes as it's not even a problem; it's browsers working as intended. When you visit different domain names, you should expect that your browser won't be aware of data (cookies) stored by other domains.