No, we don't care about chemical species, because an ideal gas is composed by point particles which do not interact except when they collide.
Basically, we are completely disregarding the structure of the atoms/molecules and their interaction potential, setting it to $0$ (a pretty heavy approximation!), so of course we won't be able to distinguish between different chemical species. In fact, as you correctly remarked, the ideal gas equation doesn't care at all about chemical species.
The difference starts to manifest itself when we consider more realistic models, such as the van der Waals gas. In the vdW approximation the molecules are approximated as hard spheres, not as point particles, so they occupy a certain amount of volume, called the excluded volume. Furthermore, we introduce an average short-ranged attractive potential between the particles, which results in an overall decrease in pressure, because it tries to keep the particles which are near the walls of the container close to each other.
The resulting equation of state is
$$
\left(p + \frac {n^2 a} {V^2}\right) (V-nb) = nRT
$$
where $a$ and $b$ are fitting parameters which are different for different species (check it out).