Nothing should happen automatically for reasons I'll get to in a minute, but if, after reviewing the question in both contexts, you believe your answer applies to both without modification, I don't see why you shouldn't post it on both. Each site is its own community and we shouldn't be evaluating questions in the context of other sites, just this one. If your answer fits here, who cares if it also fits there? Go ahead and post it. (It might not be a bad idea to mention somewhere that you're doing this, to fend off cries of "you cross-posted! bad!".)
However, there shouldn't be a general rule (or code) that automatically cross-posts answers. This is because while the question might be the same, the question in context can still be different, such that the same answer can be valid on one site and not the other. For one example, consider any cross-post to Skeptics; they have specific citation rules. For another, consider the religion sites. We once had somebody ask the exact same question on Mi Yodeya (Judaism), Christianity, and Islam; while the sites share some biblical sources that are on-topic on all three, the axioms that inform answers are very different. An answer that cites the quran or gospels as a source of authority would not be welcome on Mi Yodeya, a quran-sourced answer wouldn't be welcome on Christianity, and an answer drawing from the talmud wouldn't be welcome on either of the others.
Migrations of answered questions can sometimes cause this problem, and that's one reason that migrating questions with answers is discouraged. At least migrations are clearly labelled, so readers understand that answers might not be spot-on. But let's not make the problem worse. A human, ideally the author, should decide if this answer from Site A works over there on Site B without modifications.