No, the Department of War is the former name of the Department of the Army and nothing else. DoD is a new creation that includes the Army, historic Department of the Navy, and the other, post-WW2, new services.

"Changing it back" is completely ahistoric.

The president has no authority to do this. Federal departments and agencies are named by Congress, and even the Republicans in Congress have shown no interest in formalizing this.

The president can't change the law itself, but he can change the name they use.

The law defines the name they should be using

It's not like there is a law that says they have to use that name on their X account or what domain to use for their website or emails.

Sure, no such law that I know of. But there's also no law that suggests that anybody else needs to refer to the Department of Defense using terms that the president and his minions just made up out of thin air. I'm also arguing that going along with them, by itself, is harmful to a democratic government.