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.