The monopole moment is the total quantity of whatever it is you're considering. For example, if you're looking at a 3-dimensional multipole expansion of the mass of the earth, the monopole moment would be the total mass. That doesn't have anything to do with treating the earth like a sphere. Instead, it's the statement that the total mass of the earth (regardless of its shape) is the same no matter how far or which direction you're looking from.
Mathematically, the monopole moment is the $n=0$ moment, so
$$\int{r^n \rho(r) \;d^3(r)} = \int{r^0 \rho(r) \;d^3(r)} = \int{\rho(r) \;d^3(r)} = M$$
The answer to this question gives a good way of thinking about it intuitively.
For mass, there will always be a non-zero monopole moment. But that's not true for other quantities. For example, think about the classic example of a positive charge $q$ and negative charge $-q$ separated by a distance $d$. The total net charge is zero, so the monopole moment is zero, but the dipole moment is not: it is $qd$.
You may have noticed that in this case the dipole moment is independent of any reference point. That's because it's the lowest non-zero moment. In general, moments are a function of a reference point, but the lowest non-zero moment (and only the lowest non-zero moment) is independent of where you set the reference point. So, in the earth example, the monopole moment would be independent of the reference point, but dipole moment (and quadrupole moment, etc.) would not.
