The common saying is a hold over from when STP was defined to be $\pu{273.15 K}$ and $\pu{1 atm}$. However, IUPAC changed the definition in 1982 so that $\pu{1 atm}$ became $\pu{1 bar}$. I think the main issue is a lot of educators didn't get the memo and went right along either teaching STP as $\pu{1 atm}$ or continuing with the line they were taught ("1 mole of any gas under STP occupies $\pu{22.4 L}$") without realizing it didn't hold under the new conditions.
Just as a "proof" of this working for the old definition. \begin{align} V &=\frac{nRT}{P}\\ &=\frac{\pu{1 mol} \times \pu{8.2057338 \times 10^-2 L * atm//K * mol} \times \pu{273.15 K}}{\pu{1 atm}}\\ &=\pu{22.41396 L}\\ &\approx \pu{22.4 L} \end{align}
