Calories is a unit of energy, which cannot be changed. The efficiency of using the energy coming in different forms is however different. The original article in which the study was reported points out that indeed the matter at stake is the balance between the fat consumed and the fat burned:
The most sensitive method for detecting the rate of body fat change requires calculating daily fat balance as the difference between fat intake and net fat oxidation (i.e., fat oxidation minus de novo lipogenesis) measured by indirect calorimetry while residing in a metabolic chamber. At the end of the diet periods, our study had a minimum detectable difference in daily fat balance of 220 kcal/day (or 23 g/day) and the cumulative fat loss had a minimum detectable difference of 110 g. The observed differences in fat balance and cumulative body fat loss between RC and RF diets were substantially larger than these values and were statistically significant. While the fat balance method does not determine the anatomical location of lost fat, decreased adipose tissue triglyceride likely makes up the majority. Any additional loss of ectopic fat from liver or skeletal muscle would likely be even more beneficial.
What makes me personally skeptical is comparing the fat intake with net fat oxidation, since the former does not really replace the latter. It is better to say that carbohydrates are used more efficiently, i.e., they satisfy energetic needs of the organism better than the fatty foods, and hence the energy shortage that needs to be replenished at the expense of burning the fat stocked in the body is lower, and the weight loss os lower.
This could be modeled by the simplest energy balance equation:
$$
\frac{d M(t)}{dt} = \alpha E - \beta M^{3/4},
$$
where $M(t)$ is the body mass measured in the energy units, $E$ is the energy intake (in calories), $\alpha$ is the efficiency of energy extraction, and the last term describes the energy expenditure due to metabolism, according to the Kleiber's law.
The equilibrium solution of this equation is
$$
M=\left(\frac{\alpha E}{\beta}\right)^{\frac{4}{3}}
$$
Now, if we assume that $\alpha_{fat} < \alpha_{carb}$, the fatty foods should result in lower body mass, provided the same energy intake $E$.
In principle, one could label foods by the effective caloric content relative to that of pure fat, that is by value $\alpha_{food}E/\alpha_{fat}$.
Remark:
- since the argument in the comments is about the meaning of calories, let me also point out that calories defined as "the amount of heat needed to raise the temperature of 1 kilogram of water from 0 to 1 degree Celsius" are actually kilo-calories - a rather common misuse of this word.
- I also agree with the comments pointing out the shortcomings of the study in question: the small sample size and its short duration.