According to this comment by Shadow Wizard (September 2016):
As for unified login, the reason is different code base. Area 51 is built with the older version of Stack Exchange "engine", while all other sites with the new version - so they can't be really the same or fully synchronised, as far as I can tell.
I also have a confirmed official source from June 2016 that supports this (in addition to animuson's comment below), where Adam Lear writes (see that post for full context):
... A51 is a completely separate codebase that doesn't have direct access to the database that keeps track of the account sessions. I have some ideas on how we could make the sign-in experience there more seamless, but I need to set some time free to try them out and see what happens.
So yeah, different codebase, no direct access to account sessions.