If this were a news outline writing "Department of War" I would be concerned. But in the case of the Anthropic CEO's blog post, I can understand why they are picking their fights.
It's a testament to the broken information ecosystem we're in that many people genuinely don't know this. Most will correct themselves when told. I agree with you that those who don't are not worth engaging.
Depends where you at
They want the department to fight wars. At least they’re being honest.
Declaring a war requires Congress, and fighting a war other than in response to an invasion may be illegal under US law if Congress has not exercised its power to declare war, but that doesn't prevent wars from happening it just makes it illegal (though the only actual remedy is impeachment) for the President to wage war without authorization. And, in any case, that’s largely moot because Congress has exercised that power in an open ended (in terms of when and against whom) but limited (in authorized duration of any particular action without subsequent authorization) manner via the War Powers Act, giving every President since Nixon a blank check to start wars with full legal authority and then allow Congress an opportunity to vote to pull support from forces already in combat and hope the enemy already engaged is willing to treat the war as over as the only after-the-fact constraint.