Lets conduct a thought experiment. Build a special car.
The car is special because its wheels are perfect omni wheels. There are wheels that rotate sliplessly when the car moves along $X$, but they slide without friction when the car moves along $Y$. And there are wheels that rotate sliplessly when the car moves along $Y$, but they slide without friction when the car moves along $X$. The two directions are horizontal and perpendicular to each other.
Each set of wheels has a brake, so there are two brakes. One brake allows you to bring the $X$ component of the car velocity to $0$, the other allows you to reduce the $Y$ component to $0$. Each brake affects one component only, the brakes are in this sense "orthogonal".
Now imagine you drive the car and the velocity components are $1\frac m s$ along $X$ and $1\frac m s$ along $Y$ – but you don't know it yet because the windows are deliberately covered. Your task is to measure your initial speed with respect to the room somehow, knowing how the car works and how much it weights.
Your idea: engage the first brake and measure all the heat you will get. Because the braking wheels will still slide freely along $Y$, the velocity component along $Y$ neither will interfere with this process nor itself change. You will get the energy associated with the car's movement along $X$ only.
You do this, perform calculations and the answer is $1\frac m s$ along $X$. You repeat the procedure with the other brake and the answer is $1\frac m s$ along $Y$. Both brakes are engaged, now the car is at rest.
You got the heat corresponding to $1\frac m s$ twice. Your intuition says it's the same amount of heat you'd get by braking from $2\frac m s$ to $1\frac m s$ and then to $0$. You declare your initial speed was $2\frac m s$.
Pythagoras strongly disagrees. He says your initial speed was $\sqrt 2\frac m s$. After a bit of thinking you know he's right and you change your answer. Then you realize this means that decelerating from $\sqrt 2\frac m s$ to $1\frac m s$ would give you the same amount of heat as decelerating from $1\frac m s$ to $0$. And because you expect you can get some heat by decelerating from $2\frac m s$ to $\sqrt 2\frac m s$, then you have to admit that braking from $2\frac m s$ to $1\frac m s$ would convert more kinetic energy into heat than braking from $1\frac m s$ to $0$.
Your intuition would like to see the kinetic energy proportional to the speed (absolute value of velocity). Assume the intuition is right and imagine the initial velocity as the hypotenuse of some right triangle, where two other sides are along $X$ and $Y$ (velocity components). A traditional car could convert the length of the hypotenuse to heat by just braking. Our car with omni wheels could convert the length of one component with one brake, the length of the other component with the other brake. In total we would get more energy as heat. Different directions of velocity would give us different amounts of heat, each time at least as much as the traditional car would get. And each time we would say the final kinetic energy is 0, we converted all the kinetic energy there was.
In fact (and you know it) the kinetic energy is proportional to the speed squared. A traditional car converts the squared length of the hypotenuse to heat. Our special car converts the sum of squared lengths of two other sides. By the Pythagorean theorem these values are equal. The velocity direction doesn't matter.
To connect our thought experiment to the values in question, let's imagine you want to experimentally measure how much heat you get by braking from $10\frac m s$ to $8\frac m s$; and separately from $8\frac m s$ to $6\frac m s$. You use your special car for this.
One inconvenience though: once a brake is applied, it cannot be released until the car totally stops and you unblock things from the outside.
So you cannot just accelerate to $10\frac m s$ along $X$. If you did, you would be able to decelerate to $0$, not to the desired value of $8\frac m s$.
Worry not! Our previous experiments revealed that kinetic energy (amount of heat you can get from it) doesn't depend on direction of movement. So you accelerate the car to $8\frac m s$ along $X$ and to $6\frac m s$ along $Y$. Now your speed is $10\frac m s$ and you can reduce it to $8\frac m s$ by applying one of the brakes for good. You do this and measure the heat. The movement direction has changed but it's OK, the only thing that matters is you're traveling $8\frac m s$ now.
On the second run you accelerate the car to $6\frac m s$ along $X$ and to $2\sqrt 7\frac m s$ along $Y$. The speed is $8\frac m s$. You already expect you will get less heat than in the previous run because $2\sqrt 7 < 2\sqrt 9 = 6$. You apply the proper brake, reduce the $Y$ component to $0$ and measure the heat. It's indeed less than before.
Note your current speed is $6\frac m s$ now (along $X$ only). Aren't we lucky? You can masure the heat from the other brake when you come to a total stop and confirm it's equal to the result from the previous run, when you lost the $6\frac m s$ component.