The Poynting vector $\vec S$ is an energy density with units $W/m^2$ so at an interface there is a ratio of geometrical factor that enter in the transmittance.
Explicitly, the intensity $I=\langle S\rangle \sim E_0^2$ so
the reflectance $R$ - the ratio of reflected to incident power - is
where the $\cos\theta_r$ factor is the geometrical are on the interface intercepted by a beam reflected at angle $\theta_r=\theta_i$.
Because the transmission angle is not the same as the incident angle in general, the transmittance
but now the geometrical factors $\cos\theta_t\ne \cos\theta_i$.
Thus, the Poynting vector (in $W/m^2)$ is NOT conserved at the interface because the surface area is not conserved, but the energy is conserved once the proper ratios of areas are included.
Image credit: Eugene Hecht, Optics (5th edition) Pearson, 2016