Why gravity (through mass) or waves sums?

In physics, when we study macroscopic systems, we distinguish between what we call extensive and intensive quantities. Extensive quantities count the total amount of something, so they depend on the total amount or volume of whatever macroscopic system you're looking at. Intensive quantities are rates of change that are independent of the size of the system that you're looking at.

Examples of extensive quantities are volume, mass, energy, and entropy. All else equal, the bigger the system you look at, the larger an extensive quantity will be. Examples of intensive quantities are pressure, density, temperature, and specific heat capacity. We can have two systems that have the same temperature, but if one is larger than the other, the larger system will have internal energy.

From a macroscopic point of view, mass is extensive, so to get the total mass of a system we add the masses of all its subsystems. Temperature is not extensive, because it is a rate of change telling us how one extensive quantity (entropy) changes with respect to another extensive quantity (energy) is varied$^{dagger}$. If we have a thermodynamic system composed of interacting subsystems initially at different temperatures, the subsystems will eventually come to a common temperature that maximizes the entropy of the total system. This comports with our intuition that hot and cold systems in contact will come to thermal equilibrium at at an "average" temperature between the systems.

Microscopically, we can argue from Newtonian physics that the reason that a system of three particles acts as a single mass as because the potential energy of a particle of mass $M$ in the presence of three other masses $m_1$, $m_2$, $m_3$ is begin{align} U(mathbf{R},mathbf{r_1},mathbf{r_2},mathbf{r}_3) &= - sum_{i=1}^{3} frac{GM m_i}{|mathbf{R} - mathbf{r}_i|} + U_{text{int}}(mathbf{r}_1, mathbf{r}_2,mathbf{r}_3)\ &=U_{text{ext}}({|mathbf{R}-mathbf{r}_i|}) + U_{text{int}}(mathbf{r}_1, mathbf{r}_2,mathbf{r}_3) end{align} where $U_{text{int}}$ is the potential energy of interactions between the three masses $m_i$. (We might as well take the center of mass of the $m_i$ as the origin.)

If the external mass $M$ is far away from the three masses $m_i$ (relative to the distance of each of the $m_i$ from their common center of mass) then interaction between the external mass $M$ and the others can be expanded in the multipole expansion as begin{align} U_{text{ext}} = frac{GMm_{text{tot}}}{|mathbf{R}|} + ldots, end{align} where $m_{text{tot}} = m_1 + m_2 + m_3$. This is the microscopic justification for the additivity of gravitational mass.

From the microscopic point of view, we can't say anything about the temperature, because we only define intensive quantities like temperature for macroscopic/thermodynamic systems (as rates of change of extensive quantities in the thermodynamic limit).

$^{dagger}1/T = partial S / partial U$