Unix®, Linux®, Posix®, AIX®, and GNU® are
all registered trademarks in the United States. And they all refer to different things.
(So, when people refer to "GNU Linux," they are actually abusing
two registered trademarks having two different owners! Legally speaking, you can't do that.)
As some very expensive lawyers discovered in a long legal
defeat documented at
groklaw.net, Linux® is an entirely original operating system which shares no source-code with Unix® but which re-implements (and, improves upon) much of its API and operating concepts. Of course, Linux® today has gone much beyond its original inspirations, as has Unix.® The teams responsible for both products are of course very aware of one another and cooperate closely on many things, although their efforts diverge in some areas.