No. The Klingon War in Discovery is not the same Klingon War as that in TOS.
The Federation and the Klingons have been at peace for an unknown period of time at the start of "Errand of Mercy" in the first season of Star Trek: The Original Series, an episode that has the official but not necessarily correct date of 2267.
SPOCK: Captain, we've reached the designated position for scanning the coded directive tape.
KIRK: Good. (puts it into a decoder) We both guessed right. Negotiations with the Klingon Empire are on the verge of breaking down. Starfleet Command anticipates a surprise attack. We are to proceed to Organia and take whatever steps are necessary to prevent the Klingons
So they are at peace with the Klingons but negotiations with them seem about to break down.
UHURA: Automatic all-points relay from Starfleet Command, Captain, code one.
KIRK: Well, there it is. War. We didn't want it, but we've got it.
A state of war has just been confirmed to exist with the Klingons. Possibly one side declared war, or possibly the fighting has started.
Not allowing for the time it might possibly take the subspace message to reach the Enterprise, the war begins during the first scene of "Errand of Mercy".
I don't know how long the creators of Discovery expect that the war with the Klingons in their series will last, but it can't possibly be the same war that stars a decade or so later in "Errand of Mercy".
The Klingon War in Discovery is not the same Klingon War as that in TOS.
In "Errand of Mercy" Kirk and Kor angrily discuss the causes of the war in that episode.
KOR: You can't interfere. What happens in space is not your business.
AYELBORNE: Unless both sides agree to an immediate cessation of hostilities, all your armed forces, wherever they may be, will be immediately immobilised.
KIRK: We have legitimate grievances against the Klingons. They've invaded our territory, killed our citizens. They're openly aggressive. They've boasted that they'll take over half the galaxy.
KOR: Why not? We're the stronger! You've tried to hem us in, cut off vital supplies, strangle our trade! You've been asking for war!
KIRK: You're the ones who issued the ultimatum to withdraw from the disputed areas!
KOR: They are not disputed! They're clearly ours. And now you step in with some kind of trick.
In "The Day of the Dove":
KIRK: We only wanted to stop the fighting to save us all.
MARA: We have always fought. We must. We are hunters, Captain, tracking and taking what we need. There are poor planets in the Klingon systems, we must push outward if we are to survive.
KIRK: There's another way to survive. Mutual trust and help.
These are the only discussions of the reasons for the tensions between Klingons and the Federation in the era of TOS, and the only reasons given for the "Errand of Mercy" war as far as I can remember.
In TNG "First Contact Picard says:
PICARD: It was my error, not hers. Chancellor, there is no starship mission more dangerous than that of first contact. We never know what we will face when we open the door on a new world, how we will be greeted, what exactly the dangers will be. Centuries ago, a disastrous contact with the Klingon Empire led to decades of war. It was decided then we would do surveillance before making contact. It was a controversial decision. I believe it prevented more problems than it created.
I believe these are the only statements in 20th century episodes about the origins of conflict with the Klingons.
I do not think that they give any indication that the "Errand of Mercy" war is connected with the Discovery era War or a resumption of that conflict.
No. The Klingon War in Discovery is not the same Klingon War as that in TOS, nor does it seem to have much connection to the tensions that caused the Klingon War in TOS.