I think this debate is a little too focused on semantics. Is it possible for the Federation to technically be at war with a "race" that has no government? If they are at war, is it a cold war or a normal war? Interesting questions, but ultimately irrelevant. War is a term used to describe a military force fighting another military force. The Borg are a force of billions (maybe trillions?) with the objective of genocide against the Federation. The Borg have a military. The Borg and Federation have had military conflicts. For all intents and purposes the Federation is at war with the Borg.
Is the Federation still at war with the Borg? Yes. Wars can't expire and aren't dependent on casualties. There was the Three Hundred and Thirty Five Years' War
between the Netherlands and the Isles of Scilly without a single shot fired. Not to mention the Korean War which is currently at 60 years and counting.