The common saying is a hold over from when STP *was* defined to be $273$K and $1$ atm. However, IUPAC changed the definition in 1982 so that $1$ atm became $1$ bar. I think the main issue is a lot of educators didn't get the memo and went right along either teaching STP as $1$ atm or continuing with the line they were taught ("1 mole of any gas under STP occupies 22.4L") without realizing it didn't hold under the new conditions.

Just as a "proof" of this working for the old definition. 

$$\mathrm{V=\frac{nRT}{P}}=\frac{1 mol\times8.2057338\times10^{-2}\frac{L*atm}{K*mol}\times273K}{1atm}=22.40165L\approx22.4L$$