No, but it doesn't hurt. Also Red October is possibly the final entry in the contemporary Cold War pop culture cannon. Fighting Soviets is more satisfying than "rebel groups" or "rogue states"
But Crimson Tide is not really about the external enemy, it's about the internal one (and whether there's any ethical way to perform the mission of potentially starting a nuclear war.) That's the whole point of the wardroom discussion about von Clausewitz.