Chemistry - What is the dimension of Avogadro's constant?

The dimension/unit of the Avogadro constant $N_\mathrm{A}$ is actually $1/\mathsf{N}$ or $\pu{mol-1}$. It is shown that way in your first reference as well, i.e. $$ N_\mathrm{A} = \pu{6.02214076E23 mol-1}. $$

What is "dimensionless" is the Avogadro number or the number of atoms / molecules in a single $\pu{mol}$ of that substance, sometimes written as $N$ or $N_0$.

The numerical value is the same, but the concept is different. Basically the relationship between the two is: $N_0 = (\pu{1 mol})\times N_\mathrm{A} = \pu{6.02214076E23}$

See also the answer discussed in the comment by Ed V, where the new definition of the constant is explained: Effects of Changing Avogadro's Constant.