After having cleared every cache, history, … I was able to log into meta.mathematica.stackexchange.com, alas: After having revisited mathematica.stackexchange.com, I was not logged into meta anymore. :(
This happens for every subsite-meta.
After some experimenting, I found a real workaround (and the underlying reason for this very odd behavior):
I had to disable the rules for stackexchange in the "HTTPS Everywhere" plugin by the Electronic Frontier Foundation. Now, logging into meta.mathematica.stackexchange.com works flawlessly:
The plugin itself is not the real reason behind this, but the fact, that while https://mathematica.stackexchange.com has a valid SSL certificate, https://meta.mathematica.stackexchange.com does not!
When trying to log into meta.mathematica.stackexchange.com (or any other subsite's meta), this leads to the observed problem, since HTTPS Everywhere tries to enforce usage of the encrypted connection. Possibly, there is also a bad redirect URL somewhere within the login mechanism, specifying http or https explicitly, which it shouldn't, but I did not explore this further.
Since this happens with meta..stackexchange.com, but not meta.stackexchange.com, I suppose this might be a subdomain restriction problem of the site's SSL certificate.
This is known to SE already and not a browser issue at all.