Is Avogadro's law applicable for atoms or just for molecules?

I notice that online definitions of this experimental law always say, molecules or atoms.

The problem with just calling them all "molecules" and being done with it is some are uncomfortable with using that term for unbound atoms. If you have a container of He, there are no "molecules" in it.

So when it says "molecules or atoms", it means "molecules or unbound atoms". It's not trying to say that the total number of atoms within the different molecular species matter.

The number n in the Boyle-Mariotte-Gay-Lussac gas law represents the number of moles of the considered gas. Mole is a measure of the number of distinct particles (molecules or atoms) of a substance. Avogadro's law states that the number of gas particles in a given volume of an ideal gas is the same for different ideal gases at the same pressure and temperature. It is related to the mean kinetic energy of distinct gas particles viewed as mass points. Thus Avogadro's law holds for gases consisting of molecules as well as of atoms. Examples for gases consisting of atoms are the noble gases, e.g., helium and argon.

Your problem relates to the Ideal Gas law and Kinetic Theory rather than to Avogadro's Law alone, which is a deduction from the Ideal Gas Law.

In the Ideal Gas Law $pV=NkT$ the variable $N$ refers to the number of separate particles in the sample of gas. These particles can be individual atoms (eg atoms of the gas helium $He$) or diatomic molecules (eg molecular hydrogen $H_2$) or polyatomic molecules (eg ammonia $NH_3$) or even a mixture of different types of particles (eg air which is a mixture of $N_2, O_2, Ar, CO_2$ and smaller amounts of other gases).

If the ratio $pV/T$ is a constant for two samples of gas (which defines what it means to be an ideal gas) then it is the same constant, and the two samples contain the same number of particles regardless of their composition.

In Kinetic Theory the particles are assumed to be point masses or hard spheres. Their structure does not matter, neither does their mass, as far as this equation is concerned. The key assumption (which is justified by the accuracy with which the theory applies in experiments) is that the particles exchange energy with each other indirectly, via collisions with the walls of the container, and thereby reach an equilibrium in which each particle has the same average translational kinetic energy, regardless of its mass or its internal structure.

The structure of the particles and the composition of the gas mixture do matter when you are asking about heat capacity of gases, but the Ideal Gas equation tells you nothing about that. For that you need to know about other forms beside translational KE in which energy can be stored inside the particles of gas, such as rotational and vibrational energy.

You ask about departures (error rates) from Avogadro's Law. More generally, gases depart further from the Ideal Gas law as the size of the particles increase. The 2 major corrections to the Ideal Gas law relate to the amount of space occupied by the particles, and the forces of attraction between particles. These are expressed in the parameters $b$ and $a$ respectively in the Van der Waals equation of state for real gases
$$(p+\frac{a}{V_m^2})(V_m-b)=RT$$ where $V_m$ is the volume of one mole (Avogadro's Number) of gas particles. Both parameters $b, a$ increase as the size of the particles increase, and the bigger these parameters the greater the departure from the Ideal Gas law $pV_m=RT$ and consequently also from Avogadro's Law.