On Stack Overflow, the Analytical badge was retired when the help center rolled out. However, although the site is long out of beta, the Beta badge is still on the badges page. Shouldn't the Beta badge be retired the same way?
Not exactly. The Beta badge can still be awarded on other sites (as they're created), just not this site. The difference between the Beta badge and the Analytical badge is that the features required to award the Analytical badge no longer exist, anywhere, and it is not possible to gain the Analytical badge at all now. Theoretically, it would make sense to just remove the badge, but rather than take away a bunch of bronze badges, they opted to just label it as retired and let users keep it.
Not entirely true. The Analytical badge was retired when I made this suggestion. I also requested other badges be marked as Retired as well (i.e. the Precognitive badge, as it's no longer possible to earn it). Only the Analytical badge was marked as Retired as a result of that post.