We say charge is discrete because it always has the form $q = n e$, where $n \in \mathbb Z$.
Similarly, I can say that $m = n N_a \mu$, where $n$ is number of moles, $N_a$ is avogadro's constant and $\mu$ is the mass of one atom/molecule of that element. Since $nN_a$ is the number of atoms/molecules, it will not make sense to have $\pi$ atoms or $1.5$ atoms, so $nN_a$ is a natural number.
By this mass is clearly discrete like charge. However, my textbook says the following
One of the main differences between charge and mass is that mass is continuous and charge is not.