The informal term for a racing manual gearbox that uses toothed dog couplings instead of synchronizer rings is a dog box.
In a conventional synchromesh manual transmission, synchronizer rings help match the speeds of the gear and shaft before engagement. A dog box removes that smoothing step and uses interlocking teeth, often called dogs, to lock the selected gear to the shaft. Because the teeth engage positively rather than through friction, shifts can be extremely quick.
Racing applications accept the trade-offs because speed, strength, low weight, and compactness matter more than the smoothness expected in a road car. Drivers may shift without using the clutch, provided engine speed and drivetrain load are managed correctly. The abrupt engagement can produce noise and shock, especially if the rotating parts are poorly matched.
“Dog box” describes the unsynchronized racing transmission, while “dog clutch” names the engagement component itself. “Crash box” is a related but broader term for a non-synchronized gearbox and can also refer to older sliding-mesh designs, so it is not the best single answer here.