Why is the length of the Kelvin unit of temperature equal to that of the Celsius unit?

Kelvin history

The kelvin unit was designed so that a change of $1\ \text{K}$ corresponds to a change of $1\ ^\circ\text{C}$. This makes sense because people were working in Celsius at the time. Kelvin just realized that the Celsius scale couldn't go down arbitrarily negative. It stopped at $-273.15\ ^\circ\text{C}$.

The idea was to then make a new scale, the Kelvin scale which has the same gradations as the Celsius scale (for compatibility with the existing scale) but with the property that $0\ \text{K}$ corresponds to this special $-273.15\ ^\circ\text{C}$ temperature. In other words, it is not a coincidence but rather the kelvin was historically defined so that the two scales had the same gradation.

There is a bit of confusion regarding the triple point of water ($273.16\ \text{K}$, or $0.01\ ^\circ\text{C}$) and the freezing point of water at standard pressure ($273.15\ \text{K}$ or $0.00\ ^\circ\text{C}$). Let me clarify.

The Celsius, or Centigrade, scale was historically defined as follows. $0\ ^\circ\text{C}$ was defined to be the temperature (measured by, for example, a mercury thermometer) at which water (at standard atmospheric pressure: $101\,325\ \text{Pa}$) freezes. $100\ ^\circ\text{C}$ was chosen to be the temperature (at standard pressure) at which water boiled. Thus one degree Celsius is a gradation of temperature (as measured by a mercury thermometer, for example) equivalent $\frac{1}{100}$ of the temperature difference between the freezing and boiling points of water at standard pressure.

As early as the $17^{\text{th}}$ century scientists began to understand that the Celsius scale didn't go infinitely negative. In fact, the value where the Celsius scale would stop could be calculated and measured and it was found to occur at around $-273\ ^\circ\text{C}$. It seems to me that further refinement of laboratory experiments found the temperature to be $-273.15\ ^\circ\text{C}$. That is if you started at the freezing point of water $(0\ ^\circ\text{C})$, and went down by $273.15$ of the gradations described above, you would hit absolute zero.

Ok, we still haven't rigorously defined the kelvin. In 1967 people wanted to give good definitions to the units. The freezing point of water was a bad physical reference point because it depended on the water being at atmospheric pressure. But pressure varies with the weather and elevation on Earth so different labs might calibrate their thermometers differently by this metric. However, the temperature of the triple point of water is unambiguous (at least regarding pressure) because it only occurs when the pressure is at the right value. The triple point of water occurs at $0.01\ ^\circ\text{C}$. Thus, in 1967 it was resolved to define the kelvin as $\frac{1}{273.16}$ of the temperature of the triple point of water. This sets 1) $0\ \text{K}$ to be absolute zero as desired, 2) ensures the gradations of Kelvin were referred to a decent physical reference quantity and 3) has the effect that gradations of the Kelvin scale are the exact same as gradations of the Celsius scale.

I will leave the answer here for now. See A Peruzzi 2018 J. Phys.: Conf. Ser. 1065 12011: On the redefinition of the kelvin for details on the redefinition of the kelvin which went into effect last month.

Kelvins aren't really all that natural either; or rather, they are just as arbitrary as Celsius. You need another arbitrary quantity--the Boltzmann constant--to get the temperature unit to work with the other physical units.