John Lennon said it best: "There's nothing you can change that can't be changed..." so invariance of volatility for diffusions is a fact of life that is independent of Girsanov's theorem, or any other theorem. This is because volatility is derived from the predictable compensator of quadratic variation and quadratic variation of diffusion processes is continuous therefore predictable. If you like, quadratic variation (for diffusions or more generally for all continuous semimartingales) is equal to its drift under any probability measure. In "one-period" setting the "process" is no longer continuous and hence its quadratic variation is no longer predictable. Now its predictable compensator can be different under different measures, so we are able to change variance by changing measure.
But also note that the long-run variance of a diffusion will typically change after a change of measure. Consider, for example a geometric Brownian motion such that under the new measure the BM has constant non-zero drift rate $\mu$. At a fixed horizon $T$ the variance will change from $\exp(2T)-\exp(T)$ to $\exp(2\mu T)(\exp(2T)-\exp(T)) $.